TRANSIENTLY DEVELOPING FREE AND OPPOSING JETS IN RELATION TO GAS-ASSISTED LASER EVAPORATIVE HEATING PROCESS BY GHULAM MURSHED ARSHED A Thesis Presented to the DEANSHIP OF GRADUATE STUDIES In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE In MECHANICAL ENGINEERING KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS Dhahran, Saudi Arabia RABI’-I 1424 H MAY 2003
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TRANSIENTLY DEVELOPING FREE ANDOPPOSING JETS IN RELATION TO
GAS-ASSISTED LASER EVAPORATIVEHEATING PROCESS
BY
GHULAM MURSHED ARSHED
A Thesis Presented to the
DEANSHIP OF GRADUATE STUDIES
In Partial Fulfillment of the Requirements
for the Degree
MASTER OF SCIENCE
In
MECHANICAL ENGINEERING
KING FAHD UNIVERSITY OF PETROLEUM ANDMINERALS
Dhahran, Saudi Arabia
RABI’-I 1424 HMAY 2003
KING FAHD UNIVERSITY OF PETROLEUM AND MINERALSDHAHRAN 31261, SAUDI ARABIA
DEANSHIP OF GRADUATE STUDIES
This thesis, written by GHULAM MURSHED ARSHED under the directionof his thesis advisor and approved by his thesis committee, has been presented to andaccepted by the Dean of Graduate Studies, in partial fulfillment of the requirementsfor the degree of MASTER OF SCIENCE IN MECHANICAL ENGI-NEERING .
Thesis Committee
Dr. S.Z. Shuja (Thesis Advisor)
Prof. B.S. Yilbas (Member)
Prof. M.O. Budair (Member)
Dr. Faleh A. Al-SulaimanDepartment Chairman
Prof. Osama A. JannadiDean, College of Graduate Studies
Date
Dedicated to
My Parents, Brothers, Sisters and Wife
ACKNOWLEDGEMENTS
Words cannot at all express my thankfulness to Almighty Allah, subhanahu-wa-ta
ala, the most Merciful; the most Benevolent Who blessed me with the opportunity
and courage to complete this task.
First and foremost gratitude is due to the esteemed institution, the King Fahd
University of Petroleum and Minerals for my admittance, and to its learned
faculty members for imparting quality learning and knowledge with their valuable
support and able guidance that have led my way through this point of undertaking
my research work.
My heartfelt gratitude and special thanks to my thesis advisor learned Professor
S.Z. Shuja. I am grateful to him for his consistent help, untiring guidance, constant
encouragement and precious time that he has spent with me in completing this course
of work. I do admire his exhorting style that has given me tremendous confidence and
ability to do independent research. Working with him in a friendly and motivating
environment was really a joyful and learning experience.
I must appreciate and thank Professor B.S. Yilbas for his constructive and
iv
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positive criticism, extraordinary attention and thought-provoking contribution in my
research. It was surely an honor and an exceptional learning experience to work with
him.
I am grateful to Professor M.O. Budair for his help, advice, cooperation and
comments.
I would like to acknowledge the Chairman of Mechanical Engineering Department
Dr. Faleh Al-Sulaiman for his cooperation in providing the computer lab facility.
Sincere friendship is the spice of life. I owe thanks to my graduate fellow students
from Pakistan, India and Saudi Arabia; most particularly, Iftekhar, Ovais, Zamin,
4.5 Grid independent test for pressure along the symmetry axis at r = 0 mand t =192.30 microseconds for air jet expanding into initially stagnantair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Grid independent test for velocity magnitude along the symmetry axisat r = 0 m and t = 192.30 microseconds for air jet expanding intoinitially stagnant air. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 Grid independent test for pressure along the symmetry axis at r = 0m and t = 192.30 microseconds for helium jet expanding into initiallystagnant air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.9 Grid independent test for pressure along the symmetry axis at r =0 m and t = 192.30 microseconds for helium jet opposing the steadyturbulent air jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.10 Grid independent test for axial velocity along the symmetry axis at r= 0 m and t = 192.30 microseconds for helium jet opposing the steadyturbulent air jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Sketch of the experimental set-up [41]. . . . . . . . . . . . . . . . . . 975.2 Measured fluid height and calculated velocity of the fluid inside the
time [41]. Maximum and minimum values are represented by the ver-tical bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
x
xi
5.4 Comparison of numerical predictions with the experimental data forthe case of unsteady turbulent jet entering the water tank [41]. Theerror bars are associated with the experimental error (3.5%) as indi-cated in the previous study [41]. . . . . . . . . . . . . . . . . . . . . 101
5.6 Time development of velocity vector plots for an axisymmetric tran-sient turbulent air jet close to the jet inlet-expansion region. . . . . . 105
5.7 Time development of velocity vector plots for an axisymmetric tran-sient turbulent air jet in the radially extended and axially contractedregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.8 Time development of velocity magnitude (m/s) contours for an axisym-metric transient turbulent air jet expanding into initially stagnant air. 107
5.9 Temporal variation of velocity magnitude along the jet symmetry axisat r = 0 m for air jet expanding into initially stagnant air. . . . . . . 108
5.10 Temporal variation of turbulence kinetic energy along the jet symmetryaxis at r = 0 m for air jet expanding into initially stagnant air. . . . . 110
5.11 Time development of pressure (Pa) contours for an axisymmetric tran-sient turbulent air jet expanding into initially stagnant air. . . . . . 112
5.12 Temporal variation of pressure along the jet symmetry axis at r = 0 mfor air jet expanding into initially stagnant air. . . . . . . . . . . . . 113
5.13 Temporal variation of temperature along the jet symmetry axis at r =0 m for air jet expanding into initially stagnant air. . . . . . . . . . . 114
5.14 Time development of temperature (K) contours for an axisymmetrictransient turbulent air jet expanding into initially stagnant air. . . . 115
5.15 Ratio of jet width to penetration length with time for an axisymmetrictransient turbulent air jet expanding into initially stagnant air. . . . . 117
5.16 Penetration rate of an axisymmetric transient turbulent air jet exitinginto initially stagnant air. . . . . . . . . . . . . . . . . . . . . . . . . 118
5.17 Time development of velocity vector plots for an axisymmetric tran-sient turbulent helium jet close to the jet inlet-expansion region. . . . 120
5.18 Time development of velocity vector plots for an axisymmetric tran-sient turbulent helium jet in the radially expanded and axially con-tracted region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.19 Time development of velocity magnitude (m/s) contours for an axisym-metric transient turbulent helium jet expanding into initially stagnantair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.20 Temporal variation of velocity magnitude along the jet symmetry axisat r = 0 m for helium jet expanding into initially stagnant air. . . . . 123
5.21 Temporal variation of turbulence kinetic energy along the jet symmetryaxis at r = 0 m for helium jet expanding into initially stagnant air. . 125
5.22 Time development of pressure (Pa) contours for an axisymmetric tran-sient turbulent helium jet expanding into initially stagnant air. . . . 127
xii
5.23 Temporal variation of pressure along the jet symmetry axis at r = 0 mfor helium jet expanding into initially stagnant air. . . . . . . . . . . 128
5.24 Time development of temperature (K) contours for an axisymmetrictransient turbulent helium jet expanding into initially stagnant air. . 129
5.25 Temporal variation of temperature along the jet symmetry axis at r =0 m for helium jet expanding into initially stagnant air. . . . . . . . . 130
5.26 Temporal variation of mass fraction of helium along the jet symmetryaxis at r = 0 m for helium jet expanding into initially stagnant air. . 132
5.27 Temporal variation of mass fraction of nitrogen along the jet symmetryaxis at r = 0 m for helium jet expanding into initially stagnant air. . 133
5.28 Temporal variation of mass fraction of oxygen along the jet symmetryaxis at r = 0 m for helium jet expanding into initially stagnant air. . 134
5.29 Mass fraction of helium, nitrogen and oxygen along the jet symmetryaxis at r = 0 m and t = 192.30 microseconds for helium jet expandinginto initially stagnant air. . . . . . . . . . . . . . . . . . . . . . . . . 135
5.30 Time development of mass fraction contours of helium for an axisym-metric transient turbulent helium jet expanding into initially stagnantair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.31 Time development of mass fraction contours of nitrogen for an axisym-metric transient turbulent helium jet expanding into initially stagnantair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.32 Time development of mass fraction contours of oxygen for an axisym-metric transient turbulent helium jet expanding into initially stagnantair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.33 Ratio of jet width to penetration length with time for an axisymmetrictransient turbulent helium jet expanding into initially stagnant air. . 141
5.34 Penetration rate of an axisymmetric transient turbulent helium jetexiting into initially stagnant air. . . . . . . . . . . . . . . . . . . . . 142
5.35 Time development of velocity vector plots of He-air mixture for anaxisymmetric transiently developing helium jet opposing the steadyair jet at air jet velocity of 100 m/s. . . . . . . . . . . . . . . . . . . . 147
5.36 Time development of velocity magnitude (m/s) contours of He-air mix-ture for an axisymmetric transiently developing helium jet opposing thesteady air jet at air jet velocity of 100 m/s. . . . . . . . . . . . . . . . 148
5.37 Time development of pressure (Pa) contours of He-air mixture for anaxisymmetric transiently developing helium jet opposing the steady airjet at air jet velocity of 100 m/s. . . . . . . . . . . . . . . . . . . . . 149
5.38 Time development of mass fraction contours of helium in He-air mix-ture for an axisymmetric transiently developing helium jet opposingthe steady air jet at air jet velocity of 100 m/s. . . . . . . . . . . . . 151
5.39 Time development of mass fraction contours of nitrogen in He-air mix-ture for an axisymmetric transiently developing helium jet opposingthe steady air jet at air jet velocity of 100 m/s. . . . . . . . . . . . . 152
xiii
5.40 Time development of temperature (K) contours of He-air mixture foran axisymmetric transiently developing helium jet opposing the steadyair jet at air jet velocity of 100 m/s. . . . . . . . . . . . . . . . . . . . 153
5.41 Velocity magnitude ratio with dimensionless axial distance measuredfrom the steady air jet inlet at r = 0 m. . . . . . . . . . . . . . . . . . 155
5.42 Ratio of jet width to penetration length with time for an axisymmetrictransiently developing helium jet opposing the steady air jet. . . . . . 156
5.43 Penetration rate of an axisymmetric transiently developing helium jetopposing the steady air jet. . . . . . . . . . . . . . . . . . . . . . . . 158
5.44 Momentum ratio at different air jet velocities versus square root oftime for an axisymmetric transiently developing helium jet opposingthe steady air jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.45 Velocity vector plots for four air jet velocities at 192.30 microseconds. 1615.46 Velocity magnitude (m/s) contours for four air jet velocities at 192.30
microseconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.47 Mass fraction contours of helium for four air jet velocities at 192.30
microseconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.48 Temperature (K) contours for four air jet velocities at 192.30 microsec-
onds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.49 Pressure (Pa) contours for four air jet velocities at 192.30 microseconds.1675.50 Temporal variation of turbulence kinetic energy of He-air mixture for
four air jet velocites along the jet symmetry axis at r = 0 m. . . . . . 1685.51 Temporal variation of axial velocity of He-air mixture for four air jet
velocites along the jet symmetry axis at r = 0 m. . . . . . . . . . . . 1705.52 Variation of mass fraction of helium, nitrogen and oxygen in He-air
mixture for four air jet velocites along the jet symmetry axis at r = 0m and t = 192.30 microseconds. . . . . . . . . . . . . . . . . . . . . . 171
5.53 Temporal variation of mass fraction of helium, nitrogen and oxygen inHe-air mixture along the jet symmetry axis at r = 0 m and at air jetvelocity of 100 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
List of Tables
5.1 Thermophysical properties of fluids used in the simulations. . . . . . 173
xiv
Abstract
Name: Ghulam Murshed ArshedTitle: Transiently Developing Free and Opposing Jets in Relation
to Gas-Assisted Laser Evaporative Heating ProcessMajor Field: Mechanical EngineeringDate of Degree: MAY 2003
Laser finds wide application in industry due to its precision of operation, low cost,and local processing. Since the laser machining is involved with complex physicalprocesses, the modeling of laser induced heating gives insight into the physicalprocesses involved. Moreover, model studies reduce the experimental cost and provideparametric data for heating optimization.Laser heating of solid substrate surface results in evaporated vapor jet, which emanatesfrom the surface. Depending upon the magnitude of laser beam intensity, theevaporated vapor jet develops transiently, which implies that the velocity profile of thejet varies spatially and temporally. In practical laser heating process an assisting gasjet coaxial with the laser beam impinges onto the transiently developing vapor jet. Inthe present study, a high temperature transiently developing helium jet, imitating thevapor ejection from a laser induced cavity, and an opposing steady air jet, resemblingthe assisting gas jet, are modeled numerically. Since the thermophysical propertiesof the evaporating surface are not known in the open literature, helium at hightemperature is considered as the transiently developing jet. In order to predict theflow, temperature, and species mass fraction fields, the governing equations are solvednumerically using the finite volume method. To validate the present computationalmodel, the simulation conditions are changed and the predictions are compared withthe experimental results available in the literature.It is found that the assisting air jet influences considerably the flow field in the regionclose to the transiently developing jet. In the early stage transiently developing jetexpands in the axial direction and as the time progresses radial expansion of the jetdominates due to the assisting air jet which suppresses the transiently developingjet expansion in the axial direction; in which case, a circulation cell next to theassisting air jet boundary is developed. The radial jet developed due to opposing oftransiently developing and assisting air jets behaves similar to the free jet, and thetransiently developing jet characteristics do not affect considerably the radial free jetcharacteristics.
Master of Science DegreeKing Fahd University of Petroleum and Minerals
MAY 2003
xv
xvi
خالصة الرسالة
غالم مرشد ارشد: االسمناّقاثان حر ومعاآس له ناميان و متغّيران مع الوقت بالنسبة الى غاز مساند لعملية التبخر : عنوان الرسالة
بواسطة التسخين باليز
هندسة ميكانيكية :التخصص ربيع األول هــ , 2003مايو :تاريخ الشهادة
دقة عملية التسخين ، رخصة التكلفة والقدرة على تحديد مكان العمل بواسطة الليزر، فإن له تطبيقات نظرا ل
وبما أن عملية التصنيع بواسطة الليزر تتضمن تعقيدات فيزيائية ، فإن نمذجة التسخين بواسطة . واسعة في الصناعة
ذلك فإن دراسات النمذجة تقلل من التكلفة المخبرية باإلضافة إلى . الليزر تعطي مدخل إلى فيزيائية عملية التصنيع
. للوصول إلى أفضل حالة تسخين بواسطة الليزر ) باراميترك ( وتزود الباحث بمعلومات تحت حاالت مختلفة
اعتمادا على مقدار آثافة شعاع . نتيجة إلى تسخين أساس السطح الصلب بالليزر فإن بخار نفاث ينبعث منه
. المنبعث يتغير مع الوقت وهذا يعني بأن شكل سرعة البخار النفاث تتغير مع الفراغ والزمن ثخار النفا الليزر فإن الب
يرتطم مع البخار النفاث , في التطبيقات العملية للتسخين بالليزر يوجد غاز نفاث مساعد في نفس محور شعاع الليزر
م ومتغير مع الوقت ذو درجة حرارة عالية ، يمثل البخار في الدراسة الحالية تم نمذجة غاز هليوم نفاث نا . المتغير
نظرا . آما تم نمذجة هواء نفاث منتظم مع الوقت معاآس للهليوم . المنبعث من التجويف المستحث بواسطة الليزر
لكون الخواص الفيزيائية الحرارية للسطح المتبخر غير معروفة في البحوث المنشورة ، فإن الهليوم عند درجة
لكي يمكن التنبؤ بمجاالت االنسياب والحرارة ، فإن . رارة العالية اختير ليمثل الغاز النفاث المتنامي مع الوقت الح
وإلجازة النموذج الحسابي فإن حاالت . المعادالت المتعلقة بذلك تم حلها حسابيا باستخدام طريقة التحكم بالحجم
.رنته مع النتائج العملية الموجودة في البحوثالمحاآاة تم تغييرها ونتيجة هذا التغّير تمت مقا
في هذه الدراسة وجد أن الهواء النفاث يؤثر بشكل آبير على مجاالت التدفق في المنطقة القريبة من الهليوم
في المرحلة البدائية فإن النفاث النامي يتمدد في االتجاه المحوري وآلما تقدم الوقت فإن التمدد . النفاث المتنامي
في هذه الحالة تنتج . قطري للنفاث يصبح أآثر حضورا ألن نفاث الهواء المساعد يحصر النفاث النامي المحوري ال
والنفاث القطري الناتج بسبب النفاث المتنامي مع الوقت ونفاث الهواء . خلية دوران بجانب الهواء النفاث المتنامي
بشكل معتبر على خواص رة لخواص النفاث المتنامي فهي ال تؤث المساعد ، يشبه في سلوآه النفاث الحر ، وأما بالنسب
. النفاث القطري الحر
درجة الماجستير في العلوم
جامعة الملك فهد للبترول و المعادن الظهران المملكة العربية السعودية
ربيع األول هــ , 2003مايو
Nomenclature
a coefficients used in discretized equations
c speed of sound¡ms
¢A area (m2)
bj half velocity width of the jet (m)
b constant in source term in Eq. (4.16)
CV control volume
cp, cv specific heat of a mixture at constant pressure and volume³J
Kg · K´
cpk specific heat of the kth species at constant pressure³J
Kg · K´
C various empirical constants in turbulence model
d diameter of the jet at the inlet (m)
D jet width (m)
De, w, n or s diffusion conductance times area¡Kgs
¢Dim diffusion coefficient of the ith species in the mixture
³m2
s
´ee Favre-averaged specific internal energy of a mixture
³JKg
´E total specific internal energy of a mixture
³JKg
´Ew wall roughness parameter in Eq. (3.41)
f various wall functions used in Eqs. (3.23 & 3.25)
Fj mass flux through the face ‘j’³Kg/sm2
´Fe, w, n or s mass flow rate through the face of the control volume¡
Kgs
¢
xvii
xviii
G production rate of turbulence kinetic energy¡
Kgm · s3
¢grad gradient
hk specific enthalpy of the kth species³JKg
´hok specific enthalpy of the kth species at Tref
³JKg
´eh Favre-averaged specific enthalpy of a mixture
³JKg
´h00 fluctuating component of mixture specific enthalpy
³JKg
´H total specific enthalpy of a mixture
³JKg
´Jj total flux (convection plus diffusion) across the face ‘j’³
Kg/sm2 × [φ]
´Je, w, n or s integrated total flux over the control volume face¡
Kgs× [φ]
¢k turbulence kinetic energy
³m2
s2
´kin turbulence kinetic energy at the jet inlet
³m2
s2
´lm mixing length (m)
•min mass flux at the jet inlet (inlet to control volume)
³Kg/sm2
´Mt turbulence Mach number
•MHe total exit momentum flow rate at helium jet inlet
³Kg · m/s
s
´•MAir total exit momentum flow rate at air jet inlet
³Kg · m/s
s
´n exponent in Eq. (3.30)
bn unit normal vector
Ns total number of gaseous species
PD pressure-dilatation¡
Kgm · s3
¢p time-averaged pressure of a mixture (Pa)
xix
p 0 pressure correction (Pa)
p∗ guessed pressure (Pa)
p0 fluctuating component of mixture pressure (Pa)
P Pee function in Eq. (3.45)
Pe, w, n or s Peclet number
qw wall heat flux¡Wattm2
¢r distance along the radial direction (m)
ro radius of the jet inlet in Eq.(3.30) (m)
R gas constant³
JKg · K
´R time-averaged source term in Eq. (3.2)
³Kg/sm3
´Sh time-averaged source term in Eq. (3.5)
¡Wattm3
¢So constant in source term in Eq. (4.6)
³Kg/sm3 × [φ]
´SP coefficient of source term in Eq. (4.6)
³Kg/sm3
´Sφ arbitrary time-averaged source term in Eq. (3.46)³
Kg/sm3 × [φ]
´t time (s)
Tref mixture reference temperature (= 298.15 K)
eT Favre-averaged temperature of a mixture (K)
T 00 fluctuating component of mixture temperature (K)
Tin temperature at the jet inlet (K)
Tw wall temperature (K)
T+p dimensionless temperature at near wall point yp
Tp temperature at near wall point yp (K)
xx
eu Favre-averaged axial velocity (ms)
eun−1 velocity along x-direction of the previous iteration (ms)
eu0 velocity correction along x-direction (ms)
eu∗ guessed velocity along x-direction¡ms
¢u00 fluctuating component of axial velocity (m
s)
ui, uj arbitrary velocity (ms)
eui, euj Favre-averaged arbitrary velocity (ms)
u00i , u00j fluctuating component of arbitrary velocity (m
s)
up resultant tangential velocity (ms)
u+ dimensionless resultant tangential velocity
uo maximum axial velocity at the jet inlet (ms)
uin axial velocity at the jet inlet (inlet to control volume) (ms)
uτ resultant friction velocity (ms)
ev Favre-averaged radial velocity (ms)
evn−1 velocity along r-direction of the previous iteration (ms)
ev0 velocity correction along r-direction (ms)
ev∗ guessed velocity along r-direction (ms)
v00 fluctuating component of radial velocity (ms)
vin radial velocity at the jet inlet (inlet to control volume) (ms)
eV Favre-averaged velocity magnitude (ms)
x axial distance (m)
xi, xj arbitrary distance (m)
yp normal distance from point p to the solid wall (m)
xxi
y+p dimensionless normal distance from point p to the solid wall
eYi, eYk arbitrary Favre-averaged mass fraction
Y 00i , Y00k fluctuating component of arbitrary species mass fraction
eYHe Favre-averaged mass fraction of helium
eYN2 Favre-averaged mass fraction of nitrogen
eYO2 Favre-averaged mass fraction of oxygen
Zt penetration depth/length (m)
xxii
Greek symbols
α closure constants in Eq. (3.20)
αeu,αev,αp under-relaxation factors
γ specific heat ratio of a mixture (cp/cv)
Γ diffusion coefficient¡Kgm · s
¢δij Kronecker delta
² dissipation rate of turbulence kinetic energy³m2
s3
´²in dissipation rate of turbulence kinetic energy at the jet inlet³
m2
s3
´κ von Karman’s constant
µ laminar dynamic viscosity of a mixture¡Kgm · s
¢µt eddy viscosity of a mixture
¡Kgm · s
¢ρ time-averaged density of a mixture
¡Kgm3
¢ρ0 fluctuating component of mixture density
¡Kgm3
¢σ laminar Prandtl number
σt, σ eY turbulent Prandtl number and turbulent Schmidt number
σk, σ² turbulence constants in Eqs. (3.17) & (3.18) respectively
τ ij time-averaged stress tensor (Pa)
τw wall shear stress (Pa)
φ arbitrary variable
[φ] unit of arbitrary variable (φ)
∀ volume (m3)
xxiii
subscripts
amb ambient
in inlet
i, j arbitrary direction
i, j, k indices used to represent different species
i, j indices used in grid staggering
I, J indices used in grid staggering
P a typical node in the computational domain
o maximum
t turbulent
w wall
N,S,E,W nodes around a control volume
n, s, e, w interface of a node to its north, south, east, or west
Chapter 1
INTRODUCTION
The word laser is an acronym for “Light Amplification by Stimulated Emission of
Radiation”. Albert Einstien in 1917 showed the process of stimulated emission must
exist but it was not until 1960 that T.H. Maiman first achieved laser action at optical
frequencies in ruby. The basic principles and construction of a laser are relatively
straightforward and is somewhat surprising that the invention of the laser was so
long delayed.
In the time which has elapsed since Maiman first demonstrated laser action in
ruby in 1960, the applications of lasers have multiplied to such an extent that almost
all aspects of our daily life are touched upon by lasers. They are used in many types of
n = 2× 107 × t2 − 7693.1× t+ 0.9473 0µs ≤ t ≤ 376.92µs (3.33)
59
In order to resemble the high temperature of the evaporating surface, the emerging
jet temperature is considered as 1500 K and remains constant, i.e.,
eT = Tin (1500 K)Since the jet at the inlet to control volume consists of helium only, the mass
fraction of helium is given as:
eYHe = 1.0Values of k and ² are not known at the inlet, but some reasonable assumptions
can be made. Applying the assumption of local equilibrium (rates of production and
dissipation are both in balance) at the inlet gives the following relationships [76, 79]:
kin =1
Cµl2m
µ∂uin∂r
¶(3.34)
²in = C1/2µ k
¯̄̄̄∂uin∂r
¯̄̄̄(3.35)
Where lm is the mixing length and is given by the following Nikuradse Formula [76,79]:
lmro= 0.14− 0.08
µr
ro
¶2− 0.06
µr
ro
¶40 ≤ r ≤ ro (3.36)
The radial velocity component (vin) at this inlet is set to zero.
60
3.4.1.2 Steady Air Jet Inlet (Inlet 2)
The assisting gas jet impinging onto the transiently developing jet, resembling
the laser produced vapor jet, is a steady jet and obeys 17power law. Therefore. the
mass flux at this inlet is given by:
•min= ρuin = ρuo
µ1− r
ro
¶1/70 ≤ r ≤ ro (3.37)
Where uo and ro are independent of time at this inlet.
The assisting gas jet temperature is considered as 300 K and remains constant,
i.e.,
eT = Tin (300 K)Since the assisting gas jet is forcing against the transient jet resembling the vapor
jet, the helium content will hardly reach there at inlet 2 and hence the following
conditions for mass fraction are satisfied:
eYN2 = 0.77; & eYO2 = 0.23Values of k and ² are not known at the inlet, making similar assumptions as before
61
one can write:
kin =1
Cµl2m
µ∂uin∂r
¶(3.38)
²in = C1/2µ k
¯̄̄̄∂uin∂r
¯̄̄̄(3.39)
Where lm is the mixing length and is given by the following Nikuradse Formula:
lmro= 0.14− 0.08
µr
ro
¶2− 0.06
µr
ro
¶40 ≤ r ≤ ro (3.40)
The radial velocity component (vin) at this inlet is set to zero.
3.4.2 Outlet (Pressure Boundary)
It is considered that the flow extends over a sufficiently long domain so that;
p = pamb; eT = Tamb; eYN2 = 0.77; & eYO2 = 0.23Zero values of turbulence properties (k and ²) and mean velocities (eu & ev) are
used.
3.4.3 Entrainment Boundary
The pressure and temperature at this boundary are the same as ambient con-
dition. Zero values of turbulence properties (k and ²) and mean-velocity gradients
62
³∂eui∂xj
´are used [7, 75]. However, the boundary is far enough to satisfy the following
conditions for mass fraction:
eYN2 = 0.77; & eYO2 = 0.23
3.4.4 Symmetry Axis
Here the radial derivatives for all mean variables¡∂φ∂r
¢except the radial velocity
(ev = 0) are set to zero. Also ∂k∂rand ∂²
∂rand are set to zero at the symmetry axis [7,75].
3.4.5 Solid Wall
3.4.5.1 The Standard k − ² Model
At a solid boundary the no-slip condition applies so that both mean and fluctu-
ating velocities (u, v, u00, v00) as well as fluctuating temperature and fluctuating mass
fraction (T 00, Y 00k ) are zero but the dissipation rate (²) is finite. The equations need to
be integrated through the viscous sublayer when the boundary conditions are specified
at the wall. However, this process requires many grid points in the viscous sublayer
because the velocity gradients are very sharp here and this means additional com-
putational load. Furthermore, since the Eqs. (3.17) & (3.18) assume high Reynolds
number (because the laminar viscosity (µ) is neglected from these equations), they
are not applicable in the viscous sublayer where µ is not insignificant. Therefore, the
Universal Law of the Wall is introduced to avoid integration in the viscous sublayer.
63
This law connects the wall conditions such as wall shear stress and heat flux, and
temperature to the dependent variables just outside the viscous sublayer. This law
gives the following logarithmic relationship between the resultant tangential velocity
(up) and the dimensionless normal distance (y+p ) from point p to the solid wall:
up =uτκLn(Ewy
+p ) 30 < y+p < 500 (3.41)
Where κ (= 0.41) is von Karman’s constant, Ew (= 9.8 for smooth wall) is the wall
roughness parameter and uτ are the resultant friction velocity. uτ and y+p are given
in their respective order by the following relations:
uτ =
µτwρ
¶1/2(3.42)
y+p =ρτwypµ
(3.43)
Where yp is the normal distance from point p to the solid wall and τw is the wall
shear stress.
In addition, measurements of turbulence kinetic energy budgets indicate that the
production of turbulence kinetic energy is equal to the dissipation in the log-law
region (local equilibrium). Using this assumption and eddy-viscosity equation, Eq.
(3.16), one can develop the following wall functions [75]:
k =u2τpCµ
, ² =u3τκyp
(3.44)
64
For heat transfer the universal near wall temperature distribution at high Reynolds
number is used [75]:
T+p = −(Tp − Tw)cpρuτ
qw= σt
·u+ + P
µσ
σt
¶¸(3.45)
Where T+p is the dimensionless temperature at near wall point yp, Tp is the tem-
perature at near wall point yp, Tw is the wall temperature, qw is the wall heat flux,
u+³= up
uτ
´is dimensionless resultant tangential velocity and P is the Pee-Function,
a correction function dependent on the ratio of laminar to turbulent Prandtl num-
bers [75]. Similar to Eq. (3.45) there are laws available for species mass transfer,
relating the mass flux at the wall to the difference between the mass fraction at the
wall and the mass fraction just outside the viscous sublayer [80]. However, in the
present study the wall is insulated with respect to species mass transfer, i.e.,
∂eYi∂xj
= 0
3.4.5.2 Low Reynolds Number k − ² Model
At a solid boundary the no-slip condition applies so that both mean and fluctu-
ating velocities (u, v, u00, v00) as well as fluctuating temperature and fluctuating mass
fraction (T 00, Y 00k ) are zero. In the present study, the following conditions regarding
the turbulence kinetic energy, the dissipation rate of turbulence kinetic energy and
65
species mass fraction are applied:
k = 0;∂²
∂xj= 0; &
∂eYi∂xj
= 0
The solid wall is assumed to remain at constant temperature (Tw = 400 K) with no
radiation losses taken into account in the simulation. This condition is valid for both
high and low Reynolds number models.
Boundary conditions associated with species mass fraction are not applicable to
the transient air jet expanding into an initially stagnant air.
3.5 Initial Conditions
The initial conditions are imposed before the air/helium jet emerges into the
control volume. Therefore, initially stagnant air at ambient temperature (300 K)
and pressure (atmospheric pressure) is considered in the control volume. Moreover,
the ambient air has no helium content initially.
3.6 Properties
The density of air is considered to vary according to the ideal gas law depending
on the local pressure and temperature. The compressibility effect is accommodated
during the simulations.
66
3.7 General Form of the Differential Equations
The general form of transport equation governing the flow examined herein is
compactly represented by the following non-linear partial differential equation:
∂
∂t(ρφ) +
∂
∂xj
µρeujφ− Γφ
∂φ
∂xj
¶= Sφ (3.46)
The purpose of writing all the transport equations in one compact form is to
provide ease in numerical computation; in this case, only one transport equation
rather than all will be sufficient to deal with while performing computation.
Chapter 4
NUMERICAL METHOD AND
ALGORITHM
4.1 Introduction
In the previous chapter the mathematical modeling of turbulence to simulate the
physics of the evaporation process in non-conduction limited laser heating process
is described. This modeling process results in partial differential equations that do
not yield an analytical solution due to mathematical complexities involved and hence
some other methods of solution are required. To solve these equations numerical
methods can be employed, which are able to handle the problem of any degree of
complexity. A preliminary idea about the task of a numerical method can be obtained
by considering a heat flow situation. A number of grids is drawn to cover the whole
domain. With a sufficiently fine grid distribution, the complete distribution of the
temperature can be expressed in terms of its values at neighboring grid points. Thus
the task of the numerical method is to evaluate temperature at each grid point.
In a numerical scheme, a set of algebraic equations is derived from the governing
differential equations for the grid-point values of the temperature. The detail and
67
68
accuracy of the answer obtained depend mainly upon the proper selection of grids
and time increments. But detail and accuracy somehow require computational effort
(calculation time and computer memory). Hence, in developing a numerical scheme,
the primary consideration is a trade-off between model detail and computational
effort.
4.2 Numerical Method
Several techniques of numerical analysis exist. Among them most famous are
finite difference, finite volume, finite element, spectral and pseudo-spectral methods.
The finite volume technique is used in the present simulation for its simplicity and
accuracy [81]. Before proceeding to the finite volume method, it is appropriate to
define basic properties of numerical solutions that determine their level of accuracy.
These properties include:
• Convergence
• Consistency
• Stability
Convergence is the property of a numerical method to produce a solution which
approaches the exact solution as the grid spacing, control volume size or element size
is reduced to zero.
Consistency is the property of a numerical method to produce systems of algebraic
equations which can be demonstrated to be equivalent to the original governing partial
differential equations as the grid spacing tends to zero.
69
Stability is associated with the growth or damping of errors as the numerical
method proceeds and hence it describes whether or not the dependent variable is
bounded. For transient analysis, the dependent variable is unstable if the solution
oscillates with an amplitude that increases with time. If a technique is not stable
even round-off errors in the initial data can cause wild oscillations or divergence.
Convergence is usually very difficult to establish theoretically and in practice
Lax’s Theorem is used, which states that for linear problems a necessary and sufficient
condition for convergence is that the method is both consistent and stable. In CFD
methods this theorem is of limited use since the governing equations are non-linear.
In such problems consistency and stability are necessary conditions for convergence,
but not sufficient.
The inability to prove conclusively that a numerical solution scheme is convergent
is perhaps somewhat unsatisfying from a theoretical standpoint, but there is no need
to be too concerned since the process of making the mesh spacing very close to zero is
not feasible on computing machines with a finite representation of numbers. Round-off
errors would swamp the solution long before a grid spacing of zero is actually reached.
In CFD, there is a need of codes that produce physically realistic results with good
accuracy in simulations with finite (sometimes quite coarse) grids. Patankar [81] has
formulated rules which yield robust finite volume calculation schemes. The three
crucial properties of robust methods include:
• Conservativeness
• Boundedness
70
• Transportiveness
Conservativeness is the property of a numerical scheme which is associated with
the consistent expressions for fluxes of the fluid property through the cell faces of
adjacent control volumes.
Boundedness is akin to stability and requires that in a linear problem without
sources the solution is bounded by the maximum and minimum boundary values of the
flow variable. Boundedness can be achieved by placing restrictions on the magnitude
and signs of the coefficients of the algebraic equations. Although flow problems are
non-linear it is important to study the boundedness of a finite volume scheme for
closely related but linear, problems.
Finally all flow processes contain effects due to convection and diffusion. In
diffusive phenomena, such as heat conduction, a change of temperature at one location
affects the temperature in more or less equal measure in all directions around it.
Convective phenomena involve influencing exclusively in the flow direction so that a
point only experiences effects due to changes at upstream locations. Transportiveness
must account for the directionality of influencing in terms of the relative strength of
diffusion to convection.
Conservativeness, boundedness and transportiveness are now commonly accepted
as alternatives for the more mathematically rigorous concepts of convergence, consis-
tency and stability [75].
71
4.3 The Finite Volume Method
In this method, the calculation domain is divided into a number of non-overlapping
control volumes such that there is one control volume surrounding each grid point.
The differential equation is integrated over each control volume. Profiles (such as
step-wise and piecewise-linear profiles), expressing the variation of field variable (tem-
perature, pressure, velocity, species mass fraction, etc.) between the grid points, are
used to evaluate the required integrals. The result is the discretization equation
containing the values of field variable for a group of grid points. The discretization
equation thus obtained in this manner expresses the conservation principle of the field
variable for the finite control volume, just as the differential equation expresses it for
an infinitesimal control volume.
4.3.1 Discretization
The finite difference counterpart of the general partial differential equation (3.46)
is derived by supposing that each variable is enclosed in its own control volume and
then by integrating the partial differential equation (3.46) over the control with some
suitable assumption of field-variable profile within the control volume.
For the purpose of solution the flow domain is overlaid with a number of grids
whose center points or nodes denote the location at which all variables except ve-
locities are calculated. The latter are computed at locations midway between the
two pressure points. Thus the normal velocity components are directly available at
72
the control volume faces, where they are needed for the scalar transport -convection-
diffusion-computations. The nodes of a typical grid cluster for two dimensions are
labeled as P, N, S, E, and W. This is shown in Figure 4.1.
The integration of each term in Eq. (3.46) can be obtained with reference to the
control volume for a typical node P with its four nearest neighbors, N, S, E, and W,
in the spatial domain and Po in the time domain. The integration yields
Z t+∆t
t
½ZCV
µ∂
∂t(ρφ)− Sφ
¶d∀+
ICS
[bn. (ρeujφ− Γφgradφ)] dA
¾dt = 0 (4.1)
Divergence theorem gives
Z t+∆t
t
½ZCV
µ∂
∂t(ρφ)− Sφ
¶d∀+
ZCV
·∂
∂xj
µρeujφ− Γφ
∂φ
∂xj
¶¸d∀¾dt = 0 (4.2)
or
Z t+∆t
t
(µ∂
∂t(ρφ)− Sφ
¶∆∀+
·µρeujφ− Γφ
∂φ
∂xj
¶∆Aj
¸L2L1
)dt = 0 (4.3)
or
Z t+∆t
t
½µ∂
∂t(ρφ)− Sφ
¶∆∀+ [Jj∆Aj ]L2L1
¾dt = 0 (4.4)
Where L1 denotes w or s, L2 e or n and Jj the total flux (convection plus diffusion)
across the face ‘j’.
73
P W E
N
S
n
w e
s
∆x
∆r
Control volume
x
r
Figure 4.1: Control volume for the two-dimensional situation.
74
If Sφ is independent of time then Eq. (4.4) becomes
·ρPφP − ρoPφ
oP
∆t− Sφ
¸∆∀+ Je − Jw + Jn − Js = 0 (4.5)
Where Je, Jw, Jn, and Js are the integrated total fluxes over the control volume faces;
i.e., Je stands for Jx∆Ax over the interface e, and so on. Where the superscript ‘o’ is
used for old values (i.e., the values at previous time step).
The linearization of the source term gives
Sφ = So + SPφP (4.6)
Now Eq. (4.5) becomes
µρPφP − ρoPφ
oP
∆t
¶∆∀+ Je − Jw + Jn − Js = (So + SPφP )∆∀ (4.7)
Now integration of the continuity equation (3.1) in a similar manner to Eq. (3.46)
gives
Z t+∆t
t
½ZCV
·∂ρ
∂t+
∂
∂xj(ρeuj)¸ d∀¾ dt = 0 (4.8)
or
Z t+∆t
t
½∂ρ
∂t∆∀+ [ρeuj∆Aj]L2L1¾ dt = 0 (4.9)
75
or
Z t+∆t
t
½∂ρ
∂t∆∀+ [Fj∆Aj ]L2L1
¾dt = 0 (4.10)
or
µρP − ρoP
∆t
¶∆∀+ Fe − Fw + Fn − Fs = 0 (4.11)
Where Fj is the mass flux through the face ‘j’; Fe, Fw, Fn, and Fs are the mass
flow rates through the faces of the control volume; i.e., Fe stands for Fx∆Ax over the
interface e, and so on.
Multiplying Eq. (4.11) by φP , subtracting the resulting equation from Eq. (4.7)
and knowing that;
Je − FeφP = aE(φP − φE) (4.12)
Jw − FwφP = aW (φW − φP ) (4.13)
Jn − FnφP = aN(φP − φN) (4.14)
Js − FsφP = aS(φS − φP ) (4.15)
one can develop the following algebraic equation [81]:
aPφP = aEφE + aWφW + aNφN + aSφS + b (4.16)
76
Where
aP = aE + aW + aN + aS + aoP − SP∆∀ (4.17)
aoP =ρoP∆∀∆t
(4.18)
b = aoPφoP + So∆∀ (4.19)
aE = DeA(|Pe|) + [[−Fe, 0]] (4.20)
aW = DwA(|Pw|) + [[Fw, 0]] (4.21)
aN = DnA(|Pn|) + [[−Fn, 0]] (4.22)
aS = DsA(|Ps|) + [[Fs, 0]] (4.23)
and Pe, Pw, Pn, and Ps are the Peclet numbers: i.e., Pe stands forFeDeand so on; De,
Dw, Dn, and Ds are the diffusion conductances; i.e., De stands forΓe∆y(δx)e
and so on.
The values of A(|P |) are given in [81] for different schemes. In the present study, first
order upwind scheme is employed for which A(|P |) is unity.
Equation (4.16) is written for each of the variables, eu, ev, k, ², eρ, eYi and eT at everycell. Although the control volumes adjacent to the boundary are treated differently
from the interior ones and need different algebraic formulation, it is possible to have a
unified formulation to calculate the field variable in the near boundary region through
the use of source term [75].
77
4.4 Computation of the Flow Field
The solution of the general transport equation (3.46) presents two new problems:
• The convective term of Eq. (3.46) contains non-linear inertia terms.
• The continuity, momentum, energy, species and turbulence equations, repre-
sented by Eq. (3.46), are intricately coupled because every velocity component ap-
pearing in each equation. The most complex issue to resolve is the role played by
pressure. It appears in the momentum equations, but there is evidently no transport
equation for pressure.
If the pressure gradient is known, the process of obtaining and solving discretized
equations for velocities from momentum equations is similar to that for any other
scalar (e.g. temperature, species mass fraction, etc.) and developed schemes such as
central differencing, upwind, hybrid schemes, etc. are applicable. In general purpose
flow computations the pressure field is calculated as a part of the solution so its
gradient is normally not known beforehand. If the flow is compressible the continuity
equation may be used as a transport equation for density, and the pressure may be
obtained from the density and temperature by using the equation of state. However,
if the flow is incompressible the density is constant and hence by definition not linked
to the pressure. In this case coupling between pressure and velocity introduces a
constraint on the solution of the flow field: if the correct pressure field is applied in
the momentum equations the resulting velocity field should satisfy continuity.
Both the problems associated with the non-linearities in Eq. (3.46) and the
78
pressure velocity linkage can be resolved by adopting an iterative solution strategy
such as SIMPLE ( Semi-Implicit Method for Pressure-Linkage Equations) algorithm
of Patankar and Spalding [75].
Before outlining the algorithm it is very important to explain the grid staggering,
which is the first step to the SIMPLE algorithm. The finite volume method starts,
as always, with the discretization of the flow domain and of the general transport
equation (3.46). First there is a need to decide where to store the velocities. It
seems logical to define these at the same locations where the scalar variables, such
as pressure, temperature etc., are defined. However, if the velocities and pressure are
both defined at the nodes of an ordinary control volume, a highly non-uniform pressure
field can act like a uniform field in the discretized momentum equations [75]. For
instance, if velocities and pressure are both defined at the nodes of an ordinary control
volume and the pressure gradient terms in the momentum equations are discretized
by central differencing scheme in a uniformly discretized flow field, it is found that
all the discretized pressure terms in axial and radial directions are zero at all nodal
points even though the pressure field exhibits spatial oscillation in both directions of
a two dimensional flow field [75]. As a result, this pressure field would give the same
(zero) momentum source in the discretized equations as a uniform pressure field. This
behavior is obviously non-physical.
It is clear that, if the velocities are defined at the scalar nodes (at which scalars,
such as pressure and temperature, are defined), the influence of pressure is not prop-
erly represented in the discretized momentum equations. A remedy for this problem
79
is to use a staggered grid for the velocity components. The idea is to evaluate scalar
variables, such as pressure, density, temperature, species concentration, turbulence
kinetic energy and turbulence dissipation, at ordinary nodal points but to calculate
velocity components on staggered grids centered around the cell faces. The arrange-
ment for two-dimensional flow calculation with staggered grid arrangement is shown
in Figure 4.2. In Figure 4.2 unbroken lines (grid lines) are numbered by means of
capital letters ..., I − 1, I, I + 1, ... and ..., J − 1, J, J + 1, ... in the axial and radial
directions respectively whereas the dashed lines that construct the scalar cell faces
are denoted by lower case letters ..., i−1, i, i+1, ... and ..., j−1, j, j+1, ... in the axial
and radial directions respectively. A subscript system based on this numbering allows
us to define the locations of grid nodes and cell faces with precision. Scalar nodes,
located at the intersection of two grid lines, are identified by two capital letters: e.g.
point P in Figure 4.2 is denoted by (I, J). The axial velocities are stored at the e-
and w-cell faces of a scalar control volume. These are located at the interaction of a
line defining a cell boundary and a grid line and are, therefore, defined by a combina-
tion of a lower case letter and a capital: e.g. the w-face of the cell around point P is
identified by (i, J). For the same reasons the storage locations for the radial velocities
are combinations of a capital and a lower case letter: e.g. the s-face is given by (I, j).
The staggering of the velocity avoids the unrealistic behavior of the discretized
momentum equation for spatially oscillating pressures. A further advantage of the
staggered grid arrangement is that it generates velocities at exactly the locations
80
i i+1
E W
N
S
I, J+1
I, j
n
I, J-1
s
e w
I, j+1
i+1, J i, J
I+1, J I-1, J
I-1 I I+1
J-1
J
j
j+1
J+1
P
I, J
Figure 4.2: Staggered grid arrangement for velocity components.
81
where they are required for the scalar transport–convection-diffusion–computations.
Hence, no interpolation is needed to calculate velocities at the scalar (eg. pressure
and temperature) cell faces.
4.4.1 The SIMPLE Algorithm
The discretized momentum equations for eu an ev using Eq. (4.16) are:ai,Jeui,J = Σanbeu0nb + (pI−1,J − pI,J)Ai,J + bi,J (4.24)
As soon as the starred velocity components are obtained the pressure correction
84
equation (4.39) is solved for p 0 at all scalar nodes. Once the pressure correction field
is known, the correct pressure field may be obtained using formula (4.28) and correct
velocity components through formulae (4.31–4.34).
Afterwards, the discretization equations for the scalar variables; such as tempera-
ture, species mass fraction, and turbulence quantities are solved if they influence the
flow field through fluid properties, source terms, etc. If a particular scalar variable
does not influence the flow field, it is better to calculate it after a converged solution
for the flow field has been obtained.
If the solution is not converged the correct pressure p is treated as a new guessed
p∗ and the corrected velocity components as new guessed velocity components but
not the starred values and the whole procedure is repeated as discussed above. This
sequence of operation will be repeated over and over until a converged solution is
obtained.
The pressure correction equation is susceptible to divergence [75] unless some
under-relaxation is used during the iterative process, and new (improved) pressures
p new are obtained with
p new = p∗ + αpp0 (4.46)
Where αp is the pressure under-relaxation factor.
85
The velocities eu∗ and ev∗, and eu and ev are also under-relaxed as follows:eu∗ new = αeueu∗ + (1− αeu)eun−1 (4.47)
ev∗ new = αevev ∗+(1− αev)evn−1 (4.48)
eunew = αeueu+ (1− αeu)eun−1 (4.49)
evnew = αevev + (1− αev)evn−1 (4.50)
αeu and αev are under relaxation factors for x and y velocity components. eu∗ and ev∗are the velocity components obtained from solving the momentum equations (4.26)
and (4.27) whereas eu and ev are the corrected velocity components obtained fromvelocity correction formulae (4.31–4.34). eun−1 and evn−1 are the velocity componentsobtained in the previous iteration.
The pressure correction equation is also affected by velocity under-relaxation and
it can be shown that the d-terms of pressure correction equation (4.39) will be mul-
tiplied by the velocity under-relaxation. The second terms of the velocity correction
formulae (4.31–4.34) will also be multiplied by the velocity under-relaxation.
The whole steps followed can be well described by the flow chart (Fig. 4.3).
86
No
Initialize
Set time step
Set up time independent boundary conditions
Give initial guess to
Set up time dependent boundary conditions
Calculate the coefficients of all the discretized equations
Solve the discretized momentum Eqs. (4.26 ) & (4.27)
Solve the pressure correction Eq. (4.39)
Correct pressure by Eq. (4.28) and velocities by Eqs. (4.31 -- 4.34)
Solve the discretized Eq. (4.16) for the scalar quantities
Convergence?
Yes
Yes
No
t∆
εand,~
,~
,,~,~ kYTpvu i
εand,~
,~
,,~,~ kYTpvu i
εε =======
∆+=oo
io
ioooo kkYYTTppvvuu
ttt
and,~~
,~~
,,~~,~~Let
εand,~
,~
,,~,~ kYTpvu i
** ~,~ vu
/p
**** and,~
,~
,,~,~ εkYTpvu i
εand,~
,~
,,~,~ kYTpvu i
εand,~
,~
kYT i
Stop
Start
maxtt <
vv
uupp~~and
~~,
Set
*
**
===
Figure 4.3: The SIMPLE algorithm.
87
4.5 Grid Details and Computation
4.5.1 Free Transient Turbulent jet
Along the radial direction fine uniform grid spacing is allocated at the inlet (inlet
to control volume) while gradually increased spacing is considered away from it. Along
the axial direction grid spacing is fine near the inlet and the wall but it is gradually
increasing. The grid generated in this case is shown in Figure 4.4. The number
of grid planes used in the radial direction is 40 while 50 grid planes are used in the
axial direction, thus making a total of 2000 grid points. The grid independence test
results for pressure and velocity in case of single species are shown in Figures 4.5
& 4.6 respectively whereas these results for pressure in case of multiple species are
shown in Figures 4.7. It may be observed that for 55 × 45 grid points the results
are almost in agreement with the results of 50 × 40 grid points, i.e., the maximum
pressure and velocity magnitude differences are less than 0.1%.
Six variables in case of single species and nine variables in case of multiple species
are computed at all grid points; these are: either the two velocity components, lo-
cal pressure, two turbulence quantities and temperature or two velocity components,
three species mass fractions, local pressure, temperature and two turbulence quanti-
ties.
88
0.000
0.004
0.008
0.012
0.016
0.020
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.0000
0.0008
0.0016
0.0024
0.0000 0.0010 0.0020 0.0030 0.0040
A close-up view of the grids near inlet.
Figure 4.4: Computational domain for grid independent solution of an axisym-metric transient turbulent air/helium jet expanding into initially stag-nant air (grid size: 50x40).
89
101160
101200
101240
101280
101320
101360
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Pre
ssur
e (P
a)
40x30 Grids
45x37 Grids50x40 Grids
55x45 Grids
Figure 4.5: Grid independent test for pressure along the symmetry axis at r =0 m and t =192.30 microseconds for air jet expanding into initiallystagnant air.
90
0
20
40
60
80
100
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Vel
ocity
Mag
nitu
de (m
/s)
40x30 Grids45x37 Grids
50x45 Grids55x45 Grids
Figure 4.6: Grid independent test for velocity magnitude along the symmetry axisat r = 0 m and t = 192.30 microseconds for air jet expanding intoinitially stagnant air.
91
101160
101200
101240
101280
101320
101360
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Pre
ssur
e (P
a)
40x30 Grids
45x37 Grids50x40 Grids
55x45 Grids
Figure 4.7: Grid independent test for pressure along the symmetry axis at r = 0m and t = 192.30 microseconds for helium jet expanding into initiallystagnant air.
92
4.5.2 Unsteady Opposing Jets
Along the radial direction fine uniform grid spacing is allocated at the inlets (inlets
to control volume) while gradually increasing spacing is considered away from them so
that still there is a fine grid distribution near the wall (Wall 2 ) next to the steady air
jet inlet. Along the axial direction grid spacing is fine near inlets and the wall (Wall
1 ) next to the transiently developing jet inlet but it is gradually increasing. The grid
generated in this case is shown in Figure 4.8. The number of grid planes used in the
radial direction is 76 while 57 grid planes are used in the axial direction, thus making
a total of 3996 grid points. The grid independence test results for pressure and axial
velocity are shown in Figures 4.9 and 4.10 respectively. It may be observed that for
62 × 84 grid points the results are almost in agreement with the results of 57 × 76
grid points, i.e., the maximum pressure and axial velocity magnitude differences are
less than 0.1%.
Nine variables are computed at all grid points; these are: two velocity compo-
nents, three species mass fractions, local pressure, temperature and two turbulence
quantities.
93
Axial Distance (m)
0.0 0 0.0 1 0.0 2 0.03 0 .04
0 .00
0 .01
0 .02
0 .03
0 .04
0 .05
Aclose-upviewof the gridsnear inlets
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
A close-up view of the grids near inlets.
Rad
ial D
ista
nce
(m)
Figure 4.8: Computational domain for grid independent solution of a transientlydeveloping turbulent helium jet opposing the steady turbulent air jet(grid size: 57x76).
94
100000
102000
104000
106000
108000
0.0000 0.0004 0.0008 0.0012 0.0016 0.0020
Axial Distance (m)
Pre
ssur
e (P
a)
35x57 Grids46x68 Grids
57x76 Grids62x84 Grids
Figure 4.9: Grid independent test for pressure along the symmetry axis at r =0 m and t = 192.30 microseconds for helium jet opposing the steadyturbulent air jet.
95
-100
-50
0
50
100
0.0000 0.0005 0.0010 0.0015 0.0020
Axial Distance (m)
35x57 Grids46x68 Grids57x76 Grids62x84 Grids
Axi
al V
eloc
ity (m
/s)
Figure 4.10: Grid independent test for axial velocity along the symmetry axis atr = 0 m and t = 192.30 microseconds for helium jet opposing thesteady turbulent air jet.
Chapter 5
RESULTS AND DISCUSSIONS
5.1 Validation of the Model
Since the flow field presented in the present study resembles laser induced vapor
expansion from the cavity, there are no experimental and theoretical studies to vali-
date the present predictions. Therefore, in order to secure the validity of the present
computational model, the simulation conditions are changed in accordance with the
experiment of Kouros et al [41] to justify the capability of the model used in the
present study to predict reasonably. However, the comparison of the prediction with
the result of the above-mentioned experiment will be made after a brief look on the
experimental set-up and the conditions involved in the experiment.
In Figure 5.1 a transparent tank having dimensions 1.2 × 1.2 × 1.5 m deep is
shown, in which the jet flow field is generated by a simple apparatus consisting of a
1.2 m long round clear acrylic tube held in place by a PVC plate at one end. At the
other end, the tube is connected to a fluid feed line and a solenoid valve. The tube,
which has a 28.6 mm inner diameter, is set vertically above the tank such that the
tube exit is just below the water surface. Initially a rubber stopper is placed at the
tube exit (inside the water tank) and dyed fluid is fed into the tube from a reservoir.
96
97
Free water surfaceexposed to atmospheric
pressure
1.2 m
1.5 m
Support plate for thetube
Dimension of thetank into the shownplane is 1.2 m
0.028 m tube with dyed fluid
Tank wallWater Tank
Into the tank
1.2 m
Figure 5.1: Sketch of the experimental set-up [41].
98
Once the tube is filled, the solenoid valve is closed and the fluid is allowed to become
quiescent (1 − 2min.). The rubber stopper is then removed carefully. A minimal
amount of dye (v 5cc) diffused out of the tube prior to the start of the run. The
fluid remains in place by means similar to those which keep a liquid inside a straw
when one end is held airtight. Runs are started when the valve is rapidly opened.
Gravity forces the dyed fluid into the water tank, thus dropping the dyed fluid height
in the tube and as a result generating an unsteady velocity profile at tube exit. The
variations in instantaneous fluid height and unsteady velocity with time obtained by
Kouros et al [41] are shown in Figure 5.2. The maximum tube velocity was found
as 1.86 m/s, resulting in a jet Reynolds number (based on the tube diameter) of
5.3× 104. Hence based on this Reynolds number the jet flow field generated within
the tank is considered turbulent (the instability of the free jets takes place at any
Reynolds number [75]). Since the density difference between the jet dyed fluid and
the tank water is less than 0.1% so the buoyancy effect can be neglected.
Capturing the images of the resulting flow field using the LIF photograph, the
penetration of the tip of the jet was measured at various instants of time and is plotted
against time, which is reproduced in Figure 5.3. In order to compare the variation of
penetration length with time, the standard k − ² model is used. The comparison of
results obtained numerically with the measured values is shown in Figure 5.4. It can
be observed that both results are in good agreement. The error bars are associated
with the experimental errors, which was reported as 3.5%.
99
Figure 5.2: Measured fluid height and calculated velocity of the fluid inside thetube [41].
100
Figure 5.3: Ensembled averaged penetration length of the jet starting vortex vstime [41]. Maximum and minimum values are represented by thevertical bars.
101
0.00
0.10
0.20
0.30
0.40
0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
Measured [Kouros et al]
Predicted
Jet P
enet
ratio
n L
engt
h (m
)
Figure 5.4: Comparison of numerical predictions with the experimental data forthe case of unsteady turbulent jet entering the water tank [41]. Theerror bars are associated with the experimental error (3.5%) as indi-cated in the previous study [41].
102
5.2 Transient Jet Expansion into Stagnant Air
A transiently developing vapor jet emanating from the solid surface, which is
exposed to laser irradiation, and expanding into an initially stagnant ambient, in
practical applications the ambient is normally air, is simulated to resemble the laser
induced evaporation process. Since the actual vapor properties are not known, the
quantitative results become almost impossible. Therefore, in the first phase of the
present study two gases; air and helium, at 1500 K (imitating the evaporating tem-
perature of the laser-irradiated solid) are employed to resemble the laser-induced
vapor jet expanding into an initially stagnant air ambient. This enforces the assump-
tion that the two gas jets behave like perfect gases. Using the two different gases,
i.e., helium and air, as perfect gases may not give the complete answer; however,
it enables to demonstrate the quantitative behavior of the evaporating front (eject-
ing jet). Moreover, the evaporation of the solid surface causes the vapor jet which
develops spatially and transiently, which implies that the velocity profile of the jet
varies spatially and temporally. In the present study, this velocity profile of the vapor
jet measured previously by Yilbas et al [78] is used as jet exit (inlet to the control
volume) conditions to accommodate in the simulations. Figure 5.5 shows the jet exit
(inlet to the control volume) profiles. Furthermore, since the flow field is actually
generated by high velocity jet expanding into an initially stagnant air, therefore it is
considered turbulent. In order to accommodate turbulence, the standard k− ² model
is used.
103
microsec
0
20
40
60
80
100
120
140
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005
Radial Distance (m)
Jet A
xial
Vel
ocity
(m/s
)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.5: Profiles of jet axial velocity at the transiently developing jet inlet forvarious times [78].
104
5.2.1 Transient Air Jet into Stagnant Air
Figure 5.6 and 5.7 show velocity vectors in the region close to the jet inlet-
expansion region as well as in the radially extended and axially contracted region
at different times. The jet expansion results in the development of the circulation cell
in the region next to the jet outer surface. This is because of the flow entrainment in
this region. The orientation of the circulation cell changes as jet expansion progresses.
This is mainly because of the jet inlet velocity profile, which changes spatially with
time (Fig. 5.5). Therefore, the fluid entrainment varies with time. In the early period
(t ≤ 76.92 µs), jet expands radially more than it does axially; however, as the time
progresses axial expansion dominates over the radial expansion. This is due to: i)
the pressure which builds up close to the jet front region, and ii) the jet inlet velocity
profile which develops radially with time (Fig. 5.5). In this case, as the jet velocity
profiles become almost similar, the axial expansion of the jet is considerable.
Figure 5.8 shows the velocity magnitude contours while Figure 5.9 shows its vari-
ation along the symmetry axis as time variable. In the early period, jet expansion
is not considerable and as the time progresses jet expands radially first and then
expands further along the axial direction. This is because of the jet exit velocity
profile, which develops radially with time, as well as still air resistance opposing the
jet expansion in the axial direction. Flow entrainment is evident after t = 23.07 µs,
in this case, the outer velocity contours differ than those corresponding to earlier
time period. Although the jet penetration extends in the axial direction, velocity
105
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 15.38 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 23.07 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 76.92 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 192.30 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 307.69 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 338.46 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 361.53 microsec
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050
0.0000
0.0005
0.0010
0.0015
t = 376.92 microsec
Figure 5.6: Time development of velocity vector plots for an axisymmetric tran-sient turbulent air jet close to the jet inlet-expansion region.
106
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 376.92 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 361.53 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 307.69 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 338.46 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 192.30 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 15.38 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 23.07 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
t = 76.92 microsec
Figure 5.7: Time development of velocity vector plots for an axisymmetric tran-sient turbulent air jet in the radially extended and axially contractedregion.
107
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.8: Time development of velocity magnitude (m/s) contours for an ax-isymmetric transient turbulent air jet expanding into initially stag-nant air.
108
microsec
0
20
40
60
80
100
120
0.000 0.002 0.004 0.006 0.008
Axial Distance (m)
Vel
ocity
Mag
nitu
de (m
/s)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.9: Temporal variation of velocity magnitude along the jet symmetry axisat r = 0 m for air jet expanding into initially stagnant air.
109
magnitude reduces considerably with progressing time (Fig. 5.9). This occurs be-
cause of the jet exit velocity profile, where the mean velocity reduces significantly
(Fig. 5.5). The turbulence kinetic energy is high at the jet exit in the early period.
This can be observed from Figure 5.10, in which turbulence kinetic energy along the
symmetry axis is shown. This is because of the jet exit velocity profile. It should be
noted that the turbulence kinetic energy is associated with the jet exit velocity profile
(Eq. (3.34)). The turbulence kinetic energy reduces at jet exit; however, it attains
relatively high values along the symmetry axis during 76.92 ≤ t ≤ 192.30µs. As the
time progresses, its magnitude reduces and does not alter much along the symmetry
axis. This indicates that initial jet expansion results in high degree of turbulence and
once the jet penetration is progressed, the degree of turbulence reduces significantly.
Moreover, changes in jet exit velocity profile results in changes in velocity magnitude
along the symmetry axis. This, in turn, causes large variation in turbulence kinetic
energy, which was also observed in the previous study [82].
Figure 5.11 shows the pressure contours in the region close to the jet expansion
and Figure 5.12 shows the pressure distribution along the symmetry axis. The flow
entrainment and formation of circulation cell is evident from the pressure contours,
which is more visible after t = 76.92 µs. In the early period, jet radial expansion is
more than its axial expansion due to the pressure which builds up close to the jet
inlet region and as time reaches t = 192.30 µs axial expansion of the jet becomes
significant, in which case, the pressure in the jet front region becomes higher than
the jet inlet. This implies the shifting of the pressure peak along the symmetry axis
110
microsec
0
500
1000
1500
2000
2500
0.000 0.002 0.004 0.006 0.008
Axial Distance (m)
Tur
bule
nce
Kin
etic
Ene
rgy
( m2 /s
2 ) t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.10: Temporal variation of turbulence kinetic energy along the jet sym-metry axis at r = 0 m for air jet expanding into initially stagnantair.
111
with progressing time (Fig. 5.12). As the jet inlet profile becomes almost similar,
the pressure magnitude along the symmetry axis does not vary considerably, which
in turn results nearly similar pressure profiles along the symmetry axis.
Figure 5.13 shows temperature profiles along the symmetry axis as time variable
while Figure 5.14 shows temperature contours at different times. Temperature profiles
follow almost the profiles of velocity magnitude. This is more pronounced in the early
periods. In this case, convective heat transfer from the jet surface to its ambient is
small due to short period of time and small area of jet surface. As the time progresses,
temperature profiles extend into the jet ambient as follows the jet expansion. When
the jet exit profiles become almost similar in magnitude and shape, the extension of
the temperature profiles into the ambient becomes similar to the case observed for
the unsteady jets [29].
Figure 5.15 shows the dimensionless ratio( ratio of the jet width to penetration
depth) while the logo in the figure shows temporal behavior of penetration depth. The
penetration depth in the early period is low as compared to jet width in the radial
direction. Moreover, as the time progresses, penetration depth becomes larger and the
radial expansion of the jet becomes less than the penetration along the jet symmetry
axis. However, as the time progresses further, jet expansion results in almost self-
similar region as noted in the previous study [29]. In this case D/Zt attains almost
steady decay, which is more pronounced after 250 µs. Moreover, it was shown that
for a fixed jet exit velocity profile a self-similar transient jet could be resulted [29].
Since the gas inlet velocity profiles are varied in the present simulations (in order to
112
Rad
ial D
ista
nce
(m)
Axial Distance (m)
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(a) t= 15.38 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(b) t= 23.07 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(c) t= 76.92 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(d) t= 192.30 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(e) t= 307.69 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015( f ) t= 338.46 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(g) t= 361.53 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(h) t= 376.92 microsec
Figure 5.11: Time development of pressure (Pa) contours for an axisymmetrictransient turbulent air jet expanding into initially stagnant air.
113
microsec
100800
101200
101600
102000
102400
102800
103200
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Pre
ssur
e (P
a)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.12: Temporal variation of pressure along the jet symmetry axis at r = 0m for air jet expanding into initially stagnant air.
114
microsec
300
500
700
900
1100
1300
1500
0.000 0.002 0.004 0.006 0.008
Axial Distance (m)
Tem
pera
ture
(K)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.13: Temporal variation of temperature along the jet symmetry axis at r= 0 m for air jet expanding into initially stagnant air.
115
Rad
ial D
ista
nce
(m)
Axial Distance (m)
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(g) t= 361.53 microsec
Figure 5.14: Time development of temperature (K) contours for an axisymmetrictransient turbulent air jet expanding into initially stagnant air.
116
resemble vapor ejection from the laser-produced cavity), a self- similar region is not
observed clearly in the initial period, except when the jet exit profiles become almost
similar. This corresponds to time period of 350 µs.
Figure 5.16 shows the ratio of penetration depth (Zt) to14power of momentum
rate per unit density³ •MAir /ρ
´. It should be noted that the penetration number is
given as [29]:
Zt³ •MAir /ρ
´1/4×√t
= f(D/Zt) (5.1)
It was reported that for slow flow transient jets, the penetration number remains
constant [29]. It can be observed from Figure 5.16 that the penetration number in-
creases linearly with√t for
√t ≤ 12
õs and beyond this time it changes vastly,
i.e. constant slope of the curve indicates the constant rate of increase in penetration
number. Consequently, in the early period jet behavior is similar to that correspond-
ing to slow flow jet expansion. In this case, the penetration depth is low and the
momentum of the jet is high. As the time progresses, some portion of the jet mo-
mentum is lost due to viscous effect. This in turn reduces the specific momentum
of the jet. Moreover, as the time progresses, the jet penetration rate reduces due to
large diffusion in the region next to the jet front. Although the jet penetration rate
reduces with progressing time, the jet momentum reduces more swiftly because of
the viscous dissipation. Therefore, the rate of momentum loss is considerably higher
than the reduction in penetration rate. This, in turn, results in rapid rise of slope of
117
0.00
0.30
0.60
0.90
1.20
1.50
1.80
0 50 100 150 200 250 300 350 400
Time (ms)
D/Z
t
0.000
0.002
0.004
0.006
0.008
0 50 100 150 200 250 300 350 400
Time (ms)
Zt (
m)
Je t Penetration length of an axisymmetric transient turbulent air jet.
Figure 5.15: Ratio of jet width to penetration length with time for an axisym-metric transient turbulent air jet expanding into initially stagnantair.
118
Figure 5.16: Penetration rate of an axisymmetric transient turbulent air jet exit-ing into initially stagnant air.
119
the curve in the figure for√t ≥ 12√µs.
5.2.2 Transient Helium Jet into Stagnant Air
Figures 5.17 & 5.18 show the velocity vector in the close region of the jet expansion
as well as in the radially extended and axially contracted region at different times.
In the early times, jet expands first radially and then along the axial direction; this
is because of the jet exit velocity profile, which develops radially with time, as well
as stagnant air resistance opposing the jet expansion in the axial direction. Since the
jet inlet velocity profile changes with time, there is no specific pattern of expansion
is observed from the vector plot until after time reaches t = 76.92 µs. As the time
progresses, jet inlet velocity decays so that the jet expansion into its ambient dies.
Moreover, the flow entrainment results in a circulation cell next to the jet boundary,
which is more pronounced after t = 192.30 µs. As the time progresses the orientation
of the circular cell changes and the cell moves away from the jet boundary. As the
time progresses further, the magnitude of jet inlet velocity reduces, the size of the
circulation cell increases. In this case, jet boundary mixes with the secondary flow
generated by the circulation cell.
Figure 5.19 shows the velocity magnitude contours while Figure 5.20 shows its
variation along the symmetry axis as time variable. In the early period, jet expan-
sion is not considerable and as the time progresses jet expands radially first and then
expands further along the axial direction. This is because of the jet exit velocity
profile, which develops radially with time as well as still air resistance opposing the
120
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 15.38 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 23.07 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 76.92 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 192.30 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 307.69 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 338.46 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 361.53 microsec
0.0000 0.0010 0.0020 0.0030
0.0000
0.0005
0.0010
0.0015
t = 376.92 microsec
Figure 5.17: Time development of velocity vector plots for an axisymmetric tran-sient turbulent helium jet close to the jet inlet-expansion region.
121
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 15.38 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 23.07 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 76.92 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 192.30 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 307.69 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 338.46 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 361.53 microsec
0.0000 0.0005 0.0010 0.0015
0.0000
0.0005
0.0010
0.0015
0.0020
t = 376.92 microsec
Figure 5.18: Time development of velocity vector plots for an axisymmetric tran-sient turbulent helium jet in the radially expanded and axially con-tracted region.
122
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.19: Time development of velocity magnitude (m/s) contours for an ax-isymmetric transient turbulent helium jet expanding into initiallystagnant air.
123
microsec
0
20
40
60
80
100
120
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Vel
ocity
Mag
nitu
de (m
/s)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.20: Temporal variation of velocity magnitude along the jet symmetryaxis at r = 0 m for helium jet expanding into initially stagnant air.
124
jet expansion in the axial direction. Flow entrainment is evident after t = 76.92 µs,
in this case, the outer velocity contours differ than those corresponding to earlier
time period. Although the jet penetration extends in the axial direction, velocity
magnitude reduces considerably with progressing time (Fig. 5.20). This occurs be-
cause of the jet exit velocity profile, where the mean velocity reduces significantly
(Fig. 5.5). The turbulence kinetic energy is high at the jet exit in the early period.
This can be observed from Figure 5.21, in which turbulence kinetic energy along the
symmetry axis is shown. This is because of the jet exit velocity profile. It should be
noted that the turbulence kinetic energy is associated with the jet exit velocity profile
(Eq. (3.34)). The turbulence kinetic energy reduces at jet exit; however, it attains
relatively high values along the symmetry axis during 76.92 ≤ t ≤ 192.30µs. As the
time progresses, its magnitude reduces and does not alter much along the symmetry
axis. This indicates that initial jet expansion results in high degree of turbulence and
once the jet penetration is progressed, the degree of turbulence reduces significantly.
Moreover, changes in jet exit velocity profile results in changes in velocity magnitude
along the symmetry axis. This, in turn, causes large variation in turbulence kinetic
energy, which was also observed in the previous study [82].
Figure 5.22 shows the pressure contours in the region close to the jet expansion
and Figure 5.23 shows the pressure distribution along the symmetry axis. The flow
entrainment and formation of circulation cell is evident from the pressure contours,
which is more visible after t = 192.30 µs. In the early period, jet radial expansion
is more than its axial expansion due to the pressure which builds up close to the jet
125
microsec
0
500
1000
1500
2000
2500
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Tur
bule
nce
Kin
etic
Ene
rgy
( m2 /s
2 )
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.21: Temporal variation of turbulence kinetic energy along the jet sym-metry axis at r = 0 m for helium jet expanding into initially stagnantair.
126
inlet region and as time reaches t = 192.30 µs axial expansion of the jet becomes
significant, in which case, the pressure in the jet front region becomes higher than
the jet inlet. This implies the shifting of the pressure peak along the symmetry axis
with progressing time (Fig. 5.23). As the jet inlet profile becomes almost similar,
the pressure magnitude along the symmetry axis does not vary considerably, which
in turn results nearly similar pressure profiles along the symmetry axis.
Figure 5.24 shows temperature contours in the region of jet expansion at different
times while Figure 5.25 shows variation of temperature along the symmetry axis at
different periods. It should be noted that helium jet temperature inletting the control
volume is set to 1500 K. Consequently, as the jet expands the high temperature
helium results in convection and conduction heating of the ambient air. Moreover,
the circulation cell formed next to jet boundary enhances the convective heating of
the ambient air. This situation is evident from the temperature contours, i.e., 400 K
temperature contour extends considerably into the air ambient. As the magnitude of
the jet inlet velocity reduces, which occurs after t = 307.69 µs, the region heated by
the helium jet still remains hot, but the size of the high temperature region reduces.
This shows that the heat transfer from the high temperature jet to its ambient is
considerable despite the fact that the time is short. Moreover, due to short period
of time, the circulation cell does not convect energy from the heated ambient gas
resulting in cooling the ambient gas in this region. When examining Figure 5.25, the
temperature profiles almost follow the velocity profiles, provided that as the distance
along the symmetry axis increases, the jet front temperature reduces sharply due to
127
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(a) t= 15.38 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(b) t= 23.07 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(c) t= 76.92 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(d) t= 192.30 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(e) t= 307.69 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015( f ) t= 338.46 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(g) t= 361.53 microsec
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000
0.005
0.010
0.015(h) t= 376.92 microsec
Figure 5.22: Time development of pressure (Pa) contours for an axisymmetrictransient turbulent helium jet expanding into initially stagnant air.
128
microsec
100800
101200
101600
102000
102400
102800
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Axial Distance (m)
Pre
ssur
e (P
a)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.23: Temporal variation of pressure along the jet symmetry axis at r = 0m for helium jet expanding into initially stagnant air.
129
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.24: Time development of temperature (K) contours for an axisymmetrictransient turbulent helium jet expanding into initially stagnant air.
130
microsec
300
500
700
900
1100
1300
1500
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Tem
pera
ture
(K)
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.25: Temporal variation of temperature along the jet symmetry axis at r= 0 m for helium jet expanding into initially stagnant air.
131
heat transfer from jet to its ambient.
Figures 5.26, 5.27 & 5.28 show mass ratio of helium, nitrogen and oxygen along
the symmetry axis at different times. Depending on the jet expansion, the helium
mass ratio decays sharply along the symmetry axis. However, the deviation in helium
mass ratio at different time is because of the jet inlet velocity profile, which changes
with time (Fig. 5.5). This is particularly true for the early period (t ≤ 76.92 µs).
Figures 5.27 & 5.28 show N2 and O2 mass ratio along the symmetry axis as time
variable. The existence of N2 and O2 indicates the presence of air. The amount of
air present reduces along the symmetry axis as the time period progresses. Moreover,
at the jet inlet air does not mix with helium along the symmetry axis for all times
concerned. However, mixing of air with helium is evident as the distance along the
symmetry axis from the jet inlet increases into the downstream of the jet. This
indicates that while the helium jet expands into the stagnant air, some air molecules
remain in the region close to the jet inlet. In addition, the expansion of helium into
air accelerates the diffusional transport of air into the helium jet.
Figure 5.29 shows mass fraction of helium, nitrogen, and oxygen along the sym-
metry axis at 192.30 µs. Although the mass fraction of helium close to the helium
jet inlet is high, small fraction of nitrogen and oxygen is present there.
The same explanation regarding the enrichment of helium and depletion of nitro-
gen and oxygen along the symmetry axis can be made if mass fraction contours of
helium, nitrogen, and oxygen are considered (Figs. 5.30, 5.31 & 5.32). The only dif-
ference is that the radial expansion of the jet cannot be explained using Figures 5.26,
132
microsec
0.0
0.2
0.4
0.6
0.8
1.0
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Mas
s F
ract
ion
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.26: Temporal variation of mass fraction of helium along the jet symme-try axis at r = 0 m for helium jet expanding into initially stagnantair.
133
microsec
0.0
0.2
0.4
0.6
0.8
1.0
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Mas
s F
ract
ion
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.27: Temporal variation of mass fraction of nitrogen along the jet symme-try axis at r = 0 m for helium jet expanding into initially stagnantair.
134
microsec
0.00
0.20
0.40
0.60
0.80
1.00
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Mas
s F
ract
ion
t = 15.38
t = 23.07
t = 76.92
t =192.30
t = 307.69
t = 338.46
t =361.53
t = 376.92
Figure 5.28: Temporal variation of mass fraction of oxygen along the jet symme-try axis at r = 0 m for helium jet expanding into initially stagnantair.
135
0.0
0.2
0.4
0.6
0.8
1.0
0.0000 0.0008 0.0016 0.0024 0.0032 0.0040
Axial Distance (m)
Mas
s F
ract
ion
Helium
Nitogen
Oxygen
Figure 5.29: Mass fraction of helium, nitrogen and oxygen along the jet symmetryaxis at r = 0 m and t = 192.30 microseconds for helium jet expandinginto initially stagnant air.
136
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.30: Time development of mass fraction contours of helium for an ax-isymmetric transient turbulent helium jet expanding into initiallystagnant air.
137
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.31: Time development of mass fraction contours of nitrogen for an ax-isymmetric transient turbulent helium jet expanding into initiallystagnant air.
138
Axial Distance (m)
Rad
ial D
ista
nce
(m)
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003(a) t= 15.38 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(b) t= 23.07 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(c) t= 76.92 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(d) t= 192.30 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(e) t= 307.69 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003( f ) t= 338.46 microsec
0.000 0.002 0.004 0.006 0.008
0.000
0.001
0.002
0.003(g) t= 361.53 microsec
0.000 0.002 0.004 0.006 0.0080.000
0.001
0.002
0.003(h) t= 376.92 microsec
Figure 5.32: Time development of mass fraction contours of oxygen for an ax-isymmetric transient turbulent helium jet expanding into initiallystagnant air.
139
5.27 & 5.28. In the early period, helium jet expansion is not considerable and this
situation is evident by the sharp decay in the mass fraction of helium and sharp rise
in the mass fraction of nitrogen and oxygen close to the jet inlet along the symmetry
axis for t ≤ 23.07 µs. As the time progresses the helium jet expands radially as well as
axially but when time reaches 192.30 µs axial expansion of the jet becomes significant
as compared to the radial expansion. In the later stages of jet expansion, although
the axial expansion rate is faster than the radial expansion rate, large diffusion of
helium jet into the ambient makes the expansion rate in the axial direction slower
than the previous times of dominant axial penetration. This phenomenon enforces
the gradual decay in mass fraction of helium and gradual rise in mass fraction of
nitrogen and oxygen along the symmetry axis.
Figure 5.33 shows the ratio of jet width to penetration length (D/Zt) with time. It
should be noted that D represents the maximum width of the jet while its maximum
axial length is Zt at each time. The transient jet approaches a self-similar configu-
ration with an asymptotic D/Zt ratio. In this case, the jet approaches self-similar
as D/Zt becomes 0.65. The similar situation is observed by Hill and Ouellette [29],
provided that D/Zt obtained is less than the present finding. This is due to different
jet inlet configurations. Moreover, the transition length appears to be about 3 jet
inlet diameters, less than the data reported earlier for steady jet [83]. In the early
period (t ≤ 76.92 µs), jet is transient and it is nowhere self-similar in the flow field.
This is because of the jet inlet velocity profile, which is developing with time as shown
in Figure 5.5. As the jet inlet velocity profiles become almost similar the attainment
140
of self-similarity in the flow field becomes possible.
Figure 5.34 shows the variation of ratio of penetration length (Zt) to14power
of momentum rate per unit density³ •MHe /ρ
´with
√t. The dimensionless quantity
Ztµ •MHe/ρ
¶1/4×√t
represents the penetration number [29,32], where•MHe is the total exit
momentum flow rate. It should be noted that for the transient jet, the penetration is
a linear function of√t, i.e., Ztµ •
MHe/ρ
¶1/4×√t
= constant. It can be observed that during
√t ≤ 14√µs, the slope of the curve remains almost the same. As the time progresses,
the slope of the curve increases and varies with increasing√t. Consequently, to
achieve a constant penetration number during the jet expansion is almost impossible
in the present study, because of variation in momentum flow into the control volume
during the jet expansion. However, the slope of the curve during the period 6√µs ≤
√t ≤ 14√µs is similar to that obtained from the previous study [84].
5.2.3 Comparisons of Results Obtained Due to Transient Air
and Helium Jets
In order to compare the transient jet behavior due to air and helium jets, jet
penetration properties are considered when comparing Figures 5.15 and 5.33 in which
ratio of jet width to penetration length (D/Zt) with time are shown. Although the
behavior of (D/Zt) with time is similar, the slopes of both curves vary considerably.
In this case, the slope of transiently expanding helium jet results in higher slope
[∆(D/Zt)/∆t] than air jet. This indicates that air jet penetrates more into its ambient
141
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0 50 100 150 200 250 300 350 400
Time (ms)
D/Z
t
0.0000
0.0010
0.0020
0.0030
0.0040
0 50 100 150 200 250 300 350 400
Time (ms)
Zt (
m)
Je t Penetration length of an axisymmetric transient turbulent helium jet.
Figure 5.33: Ratio of jet width to penetration length with time for an axisymmet-ric transient turbulent helium jet expanding into initially stagnantair.
142
Figure 5.34: Penetration rate of an axisymmetric transient turbulent helium jetexiting into initially stagnant air.
143
than the helium jet. This occurs because of the density of the helium, which is
lower than the air. This can also be observed from temperature curves (Figs. 5.13
& 5.25). In the case of penetration number, both jets behave similarly (Figs.5.34
& 5.16), provided that longer expansion of the air jet results in larger gradientsh∆(Zt/ρ
•M)1/4/∆
√t)i. This, is, again due to the thermophysical properties of the
fluids, which differ considerably. It should be noted that helium has the density
almost 1/10 of air. Consequently, the momentum of helium jet onset of leaving the
solid surface is considerably smaller than that corresponding to air jet, despite the
fact that both jets have the same transiently varying velocity profiles at this location.
Moreover, the specific heat capacity of helium is almost 5 times higher than the air. It
is, therefore, expected that the convective heating of the ambient gas by the transient
jet is higher in the case of helium. Since the inlet momentum of helium jet is lower
than the air jet, this situation is not reflected in temperature contours in the region
next to helium jet (Fig. 5.24).
5.3 Opposing Jets
In practical laser heating process an assisting gas jet coaxial with the laser-heating
source impinges onto the transiently developing vapor jet emanating from the irra-
diated solid surface. In the next phase of the present study, a high temperature
transiently developing helium jet, resembling the vapor ejection from a laser induced
surface, and an opposing steady air jet, resembling the assisting gas jet, are stud-
ied. Since the thermophysical properties of the evaporating surface are not known in
144
the open literature, helium at 1500 K (imitating the evaporating temperature of the
laser-irradiated solid) is considered as the transiently developing jet. This enforces
the assumption that the transiently developing helium and steady air jets behave
like perfect gases. Using helium, which is different from the laser irradiated vapor,
and air at such a high temperature as perfect gases may not give the complete an-
swer; however, it enables to demonstrate the quantitative behavior of the opposing
jets-situation which is caused by the impingement of the assisting gas jet with the
vapor jet. Moreover, the velocity profile of the transiently developing vapor jet mea-
sured previously by Yilbas et al [78] is used as the velocity profile of the transiently
developing helium jet at one inlet to the control volume (inlet 1), and 17th− power
velocity profile is used as the velocity profile of the steady air jet (assisting gas jet)
at the second inlet to the control volume (inlet 2). Figure 5.5 shows the velocity
profiles of the transiently developing vapor jet. Furthermore, since the flow field is
actually generated by the opposing of the high velocity transiently developing helium
and steady air jet and the flow field is mainly of interest in the near wall region,
therefore low Reynolds number k−² model is used to accommodate turbulence in the
simulations. In this phase of the study, the transient effect and the influence of the
assisting gas velocity on the flow, temperature, pressure, and species mass fraction
fields are discussed.
145
5.3.1 Transient Effect on The Flow Field Due to Opposing
Jets
Opposing jet situation resembling the laser gas assisted processing is discussed.
The transiently developing jet resembling the laser-induced vapor plume and the
steady jet impinging onto the transiently developing jet are considered. The geometric
configurations of the jets are selected in accordance with the actual laser processing
conditions.
Figure 5.35 shows velocity vectors in the region close to the transiently developing
jet region for different periods. The transiently developing jet penetrates into steady
opposing jet during the early period (t ≤ 23.07 µs). As the transiently developing
jet size increases in the radial direction, the opposing jet suppresses the penetration
of the jet in the axial direction. In this case, a circulation cell next to the steady
jet boundary and close to the transient jet is developed (t = 76.92 µs). As the time
progresses further (t = 192.30 µs), radial extension of the transiently developing jet
enhances, which in turn accelerates the radial flow in the region close to the solid wall.
Consequently, the orientation of the circulation cell changes and it moves away from
the steady jet boundary. In this case, transiently developing jet velocity develops
further in the radial direction and its velocity profile enables the impinging jet to
spread radially in the region close to the impinging jet. As the time progresses further
(t ≥ 361.53 µs), the average velocity of the transiently developing jet reduces and
the impinging jet suppresses further penetration of the transiently developing jet and
146
enhanced radial flow close to the wall ensures the steady jet expansion in this region.
The radial expansion of the steady jet at t ≥ 361.53 µs is also observed from Figure
5.36, in which velocity magnitude contours are shown. The size of the circulation
cell grows as time progresses. It should be noted that the transiently developing jet
profiles at control volume inlet changes with time. Consequently, its effect on the size
and orientation of the circulation cell varies with time.
Figure 5.37 shows the pressure contours in the region close to the transiently de-
veloping jet. The high magnitude pressure contours extend into the steady jet region
along the axial direction during the early period. This indicates that a triangle like
transiently developing jet velocity profile penetrates deeper into the steady impinging
jet. As the time progresses, the transiently developing jet expands radially, which
in turn influences streamline curvature of the steady impinging jet. Consequently,
transiently developing jet penetration into the steady jet region is suppressed by
the stagnation region developed in the vicinity of transiently developing jet bound-
ary. This enhances the development of the radial flow; therefore, a circulation cell
is formed next to the steady jet boundary, provided that the pressure field gradu-
ally reduces in this region. In the case of t = 376.92 µs, the appearance of pressure
contour (101.270 KPa) may suggest the formation of the secondary cell close to the
developing jet boundary. However, close examination of velocity contours and ve-
locity vectors indicate that the flow mixing in this region occurs and formation of
secondary cell is unlikely.
Figure 5.38 shows helium mass fraction contours in the region close to the tran-
147
Rad
ialD
ista
nce
(m)
A xial D istance (m)
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
15.38 microsec 23.07 microsec 76.92 microsec
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
192.30 microsec 307.69 microsec 338.46 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
361.53 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
369.23 microsec
0.0000 0.0005 0.0010 0.0015 0.0020
376.92 microsec
Figure 5.35: Time development of velocity vector plots of He-air mixture for anaxisymmetric transiently developing helium jet opposing the steadyair jet at air jet velocity of 100 m/s.
148
A xial D istance (m)
Rad
ialD
ista
nce
(m)
6 8
15
3560
6
4
759040.000
0.001
0.002
0.003
0.004
0.005
15.38 microsec
6 8
15
356015 10
90
23.07 microsec
815 10
90
45
60
6
75
2535
45 60
76.92 microsec
88
15
45
60
75 90
6
60
10
35
25
0.000
0.001
0.002
0.003
0.004
0.005
192.30 microsec
6
815
25
45
9075
6
60
3510
45
60
307.69 microsec
6
8
25
9075
4535
6
45
1015
60
338.46 microsec
6
8
6
15
10
25
75 9045
35
45
60
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
361.53 microsec
6
15
25
60 75
35
45
810
9060
6
0.000 0.001 0.002
369.23 microsec
6
15
25
60 75
35
45
810
9060
0.000 0.001 0.002
376.92 microsec
Figure 5.36: Time development of velocity magnitude (m/s) contours of He-airmixture for an axisymmetric transiently developing helium jet op-posing the steady air jet at air jet velocity of 100 m/s.
149
Rad
ialD
ista
nce
(m)
A xial D istance (m)
102600
101830
102000
102204101190
0.000
0.001
0.002
0.003
0.004
0.005
15.38 microsec
1010
70
1012
68
102600101190
101190
101268
101140
0.000
0.001
0.002
0.003
0.004
0.005
192.30 microsec
101830
102200
102600
101485102000
23.07 microsec
101315101290
101270101315
101190
101485
102600
307.69 microsec
101315
101485101315
101290
101190
102600
101270
101360338.46 microsec
101190
101290
101290
101270
101270
101315
101640102600 101315
0.000 0.001 0.002
369.23 microsec
101190
101290
101290
101270
101270
101315
101640102600 101315
0.000 0.001 0.002
376.92 microsec
101190
10127010
1290
101315
101315
101315
101640
101315
101360
102600
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
361.53 microsec
101360
101190
101400
101315
101390
102600
76.92 microsec
Figure 5.37: Time development of pressure (Pa) contours of He-air mixture for anaxisymmetric transiently developing helium jet opposing the steadyair jet at air jet velocity of 100 m/s.
150
siently developing jet. In the early time when transiently developing jet axially ex-
tends into the steady impinging jet, helium mass fraction is higher close to the tran-
siently developing jet entering region. As the time progresses, radial expansion of
the transient jet and flow mixing in the region next to the transient jet boundary
lower the helium mass fraction while increase the nitrogen mass fraction (Fig.5.39).
Moreover, mass fraction of helium is suppressed further by the steady impinging jet
for t = 376.92 µs. In the case of nitrogen mass fraction (Fig.5.39), nitrogen mass
fraction reduces in the region close to the transient jet. This is more pronounced in
the early times. Moreover, flow mixing close to the transient jet boundary enhances
the nitrogen mass fraction in this region, which is clearly observed for t ≥ 307.69 µs.
It should be noted that the impinging steady jet is air, which composes of oxygen
and nitrogen. The presence of nitrogen indicates the presence of air in the flow field.
Figure 5.40 shows temperature contours in the region close to the transient jet.
Transiently developing helium jet is at 1500 K at the onset of emanating from the
wall (cavity wall) while steady impinging jet temperature is 300 K at the jet entry.
Consequently, transiently developing jet core temperature remains high while at the
mixing region temperature of the steady jet increases due to the convective and
conductive heat transfer in the mixing region. In the early period t = 15.38 µs,
although the transient jet expands axially into the steady jet, temperature across
the transient jet boundary does not attain high values. Moreover, transient jet core
temperature extends further into the steady jet at t = 23.07 µs. Due to radial
extension of the transiently developing jet and modification of streamline curvature
151
A xial D istance (m)
Rad
ialD
ista
nce
(m)
0.1850.300
0.450
0.065
0.00000
0.00025
0.00050
0.00075
0.00100
15.38 microsec
0.065
0.185
0.950
0.700
0.300
23.07 microsec
0.185
0.3000.4500.700
0.1250.950
76.92 microsec
0.065
0.1250.1 850 .5500.80
0
0 .37
5
0.00000
0.00025
0.00050
0.00075
0.00100
192.30 microsec
0.065
0.125
0.7 0
0 0 .3 7
5 0.185
307.69 microsec
0.7 0
0
0 . 300
0.065
0.125
0.4 5
0
0.1 85
338.46 microsec
0.065
0.185
0 .125
0.70
0
0.375
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
361.53 microsec
0.185
0.125
0.70
0
0.3 75
0.065
0.00000 0.00005 0.00010
369.23 microsec
0.185
0.125
0.70
0
0 .375
0.065
0.00000 0.00005 0.00010
376.92 microsec
Figure 5.38: Time development of mass fraction contours of helium in He-air mix-ture for an axisymmetric transiently developing helium jet opposingthe steady air jet at air jet velocity of 100 m/s.
152
Rad
ialD
ista
nce
(m)
A xial D istance (m)
0.600
0.500
0.375
0.700
0.00000
0.00025
0.00050
0.00075
0.00100
15.38 microsec
0.125 0.250
0.500
0.700
0.040
23.07 microsec
0.650
0.600
0.5000.3750.250
0.125
0.700
76.92 microsec
0.700
0.6500.600
0.50
0
0.1 2
5
0.37
5
0.00000
0.00025
0.00050
0.00075
0.00100
192.30 microsec
0.650
0.60 00 .50
0
0 .37
5
0 .1 2
5
0.700
307.69 microsec
0.700
0.600
0 .50 00.37
5
0.12
5
0.6 50
338.46 microsec
0.6500. 5000.3 7
5 0 .60 0
0.700
0.1 2
5
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
361.53 microsec
0.700
0.650
0 .60 00.1 2
5
0.37
5
0.5 00
0.00000 0.00005 0.00010
369.23 microsec
0.700
0.650
0.600
0.500
0.125
0 .375
0.00000 0.00005 0.00010
376.92 microsec
Figure 5.39: Time development of mass fraction contours of nitrogen in He-airmixture for an axisymmetric transiently developing helium jet op-posing the steady air jet at air jet velocity of 100 m/s.
153
A xial D istance (m)
Rad
ialD
ista
nce
(m)
375
375445
600
975
1275 7500.00000
0.00025
0.00050
0.00075
0.00100
15.38 microsec
375
1425
13001125
750
445
375
23.07 microsec
1 42 5
1280 1125
975
825675
600
445
400
76.92 microsec
1 42 5
1200 97 5
7 50
445
375
600525
0.00000
0.00025
0.00050
0.00075
0.00100
192.30 microsec1 2
0 0
825 5 25
445
375
675
307.69 microsec
975
7 5 0
6 00
525
445
375
1200
338.46 microsec
9 75 60 0
445525
375
1200 7 5 0
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
361.53 microsec
8 25
525
445
375
1200
0.00000 0.00005 0.00010
369.23 microsec
825
5 25445
375
1200
0.00000 0.00005 0.00010
376.92 microsec
Figure 5.40: Time development of temperature (K) contours of He-air mixturefor an axisymmetric transiently developing helium jet opposing thesteady air jet at air jet velocity of 100 m/s.
154
of the steady jet, temperature field is modified considerably with progressing time.
In this case, flow mixing enhances the temperature rise in the region close to the
transiently developing jet.
Figure 5.41 shows velocity magnitude ratio with x/bj along the symmetry axis for
different periods. It should be noted that ”bj” represents the half velocity width of
the jet and the axial distance is measured from the steady air jet inlet. Moreover, x/bj
remains constant along the symmetry axis for free jets [85]. The steady impinging jet
expands gradually along the symmetry axis, since velocity magnitude ratio remains
almost the same for x/bj ≤ 5. Due to the transiently developing jet, steady jet
expansion in axial direction is suppressed. In this case, radial expansion of steady jet
occurs as observed from Figure 5.36. This results in radial jet emanating from the
region where steady and transiently developing jets meet. Although the transiently
developing jet modifies the flow structure around the jets-meeting-region, the radial
jet behaves like almost a steady jet. Consequently, x/bj ratio remains constant with
velocity magnitude ratio as consistent with the previous work [56].
Figure 5.42 shows the ratio of jet width to penetration length (D/Zt) of the
transiently developing jet. ”D” represents the maximum width of the jet and the
maximum jet height in axial direction is Zt. It should be noted that the transient
jet approaches the self-similar situation when D/Zt remains almost constant with
time. In the present case, transiently developing jet does not approach a self-similar
situation at any time. The sharp increase in D/Zt represents the radial expansion of
the jet due to the opposing steady jet. Once the developing jet reaches a stage where
155
0.4
0.6
0.8
1.0
1.2
0 2 4 6 8
Axial Distance/Half Velocity Width
Vel
ocity
Mag
nitu
de/M
ax. A
ir J
et V
eloc
ity
t = 76.92t = 192.30t = 307.69t = 361.53t = 376.92
microsec
Figure 5.41: Velocity magnitude ratio with dimensionless axial distance measuredfrom the steady air jet inlet at r = 0 m.
156
0
5
10
15
20
25
0 50 100 150 200 250 300 350 400
Time (ms)
D/Z
t
0.00000
0.00008
0.00016
0.00024
0.00032
0.00040
0 50 100 150 200 250 300 350 400
Time (ms)
Zt (m
)
Je t Penetration length of an axisymmetric opposing transiently developing helium jet.
Figure 5.42: Ratio of jet width to penetration length with time for an axisym-metric transiently developing helium jet opposing the steady air jet.
157
the radial expansion of the jet remains almost steady; in which case, penetration of
transiently developing jet in the axial direction is suppressed by the opposing steady
jet. The decaying of D/Zt with time attributes to the decaying of the transiently
developing jet as can also be observed from Figure 5.5, i.e., the mean velocity of the
transiently developing jet decays and its magnitude reduces gradually to zero.
Figure 5.43 shows the variation of ratio of penetration length (Zt) to14power
of momentum rate per unit density³ •MHe /ρ
´with
√t. The dimensionless quantity
Ztµ •MHe/ρ
¶1/4×√t
represents the penetration number [29,32], where•MHe is the total exit
momentum flow rate at the transiently developing jet inlet. It should be noted that the
penetration number remains constant for self-developing transient jets. In the present
situation, the penetration number attains almost steady value during 10 − 15√µs.
This period corresponds to the transiently developing jet entering velocity profiles
being similar (Fig. 5.5). Moreover, sharp increase in the penetration number during
√t ≤ 5
õs represents the axial penetration of the jet, which is significant in the
early periods as can be observed from Figure 5.36. This is also evident from Figure
5.44, in which the ratio of total momentum flow rates corresponding to transiently
developing and steady jets. In this case the momentum ratio increases sharply in
the early period and increases steadily during 7.5 ≤ √t ≤ 12.5√µs. Despite the fact
that the momentum flow rate increases sharply in the early period and it is inversely
proportional to the penetration number, substantial increase in Zt enables to increase
the penetration number during the period√t ≤ 5√µs.
158
Figure 5.43: Penetration rate of an axisymmetric transiently developing heliumjet opposing the steady air jet.
159
Figure 5.44: Momentum ratio at different air jet velocities versus square root oftime for an axisymmetric transiently developing helium jet opposingthe steady air jet.
160
5.3.2 Influence of Assisting Gas Velocity on The Flow Field
Due to Opposing Jets
In order to examine the influence of the magnitude of steady jet mean velocity
on the flow field due to transiently developing opposing jet, the mean velocity of
the steady impinging jet is varied. In order to simplify the arguments, the results
obtained for the flow field at time t = 192.30 µs are presented in the figures. It
should be noted that at t = 192.30 µs, transiently developing jet profiles at the onset
of entering the control volume is almost developed in size (Fig. 5.5).
Figure 5.45 shows velocity vectors for different mean entering velocity of the
steady jet at t = 192.30 µs. In the case of low jet velocity (5 m/s) transiently devel-
oping jet expands axially into the steady jet and the radial expansion of the jet is not
considerable. Therefore, a weak circulation cell radially away from the transiently
developing jet is developed. As the mean velocity of the steady jet increases, axial
expansion of the transiently developing jet is suppressed and due to the streamline
curvature effect of the steady impinging jet, radial flow next to the transiently devel-
oping jet is developed. The strength of the circulation cell increases, which in turn
enhances the flow mixing next to the transiently developing jet boundary. Moreover,
as the mean velocity of the steady jet increases further, the orientation of the cir-
culation cell changes due to the enhancement of the radial flow entrainment. The
strength of the circulation cell increases while the radial expansion of the transiently
developing jet becomes considerable. In this case axial expansion of the transiently
161
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Airjet velocity50 m/s
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Airjet velocity100 m/s
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Airjet velocity150 m/s
0.0000 0.0005 0.0010 0.0015 0.0020
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Airjet velocity5 m/s
Figure 5.45: Velocity vector plots for four air jet velocities at 192.30 microseconds.
162
developing jet becomes minimum. This can also be seen from Figure 5.46, in which
contours of velocity magnitude are shown. The flow mixing next to the transiently
developing jet boundary is evident for 50 m/s mean velocity of the steady jet.
Figure 5.47 shows helium contours for different mean velocity of the steady jet
at 192.30 µs. In the case of low mean velocity (5 m/s), the axial expansion of
the transiently developing jet enables helium to penetrate into the steady impinging
jet. Consequently, helium mass fraction attains considerably high values along the
symmetry axis. It should be noted that steady impinging jet is air while transiently
developing jet is helium. Moreover, existing of air in the helium jet is not observed.
As the steady impinging jet velocity increases (50 m/s), helium penetration into air
along the axial direction is suppressed and mixing of the air with helium next to the
transiently developing jet boundary reduces the helium mass fraction in this region.
As the steady impinging jet velocity increases further, helium penetration into air
is suppressed considerably. The streamline curvature of the steady impinging jet
enhances the mixing of helium with air in both axial and radial directions.
Figure 5.48 shows temperature contours for different entering mean velocity of
steady jet at t = 192.30 µs. It should be noted that the helium jet is at 1500 K
while the steady impinging jet is at 300 K. Consequently, as transiently developing
jet expands into the steady jet, temperature of air increases in the region close to
the transiently developing jet boundary. This situation is clearly observed for 5 m/s
steady jet velocity. In this case, convection and conduction heating of the air take
place. When the mean velocity of the steady jet increases, flow mixing modifies the
163
Axial Distance (m)
Rad
ialD
ista
nce
(m)
1
3
4
6
8
60
25
4
3
2
275
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
Airjet velocity5 m/s
6
8
25
2515
35
35 4548
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
Airjet velocity50 m/s
6
8
8
10
15
25
35
907545
6060
45
95
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
Airjet velocity100 m/s
68
10
15
2535
4560
75
145115
9090
130
75
10
0.000 0.001 0.002
0.000
0.001
0.002
0.003
0.004
0.005
Airjet velocity150 m/s
Figure 5.46: Velocity magnitude (m/s) contours for four air jet velocities at 192.30microseconds.
164
Axial Distance (m)
Rad
ialD
ista
nce
(m)
0.185
0.125
0.375
0.550
0.650
0.800
0.9000.950
0.065
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity50 m/s
0.065
0.125
0.185
0 .2 50
0.375
0.80
0
0.95
0
0 .55
00.65
0
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity100 m/s
0.1250.185
0 .3 7
5
0.55
0
0.95
0
0.65
0 0.25 0
0.065
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity150 m/s
0.125
0.999
0.800
0.375
0.980
0.065
0.185
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity5 m/s
Figure 5.47: Mass fraction contours of helium for four air jet velocities at 192.30microseconds.
165
Axial Distance (m)
Rad
ialD
ista
nce
(m)
1400
800
600
525
445
375
1200
100 0
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity100 m/s
1 40 0
1200
1000
800
700600
525
445
375
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity150 m/s
445
1497
1495
1485
800
1200
525
1400
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity5 m/s
1425
1350
1200
1050 925850
700
600525
445
375
1400
0.00000 0.00005 0.00010
0.00000
0.00025
0.00050
0.00075
0.00100
Airjet velocity50 m/s
Figure 5.48: Temperature (K) contours for four air jet velocities at 192.30 mi-croseconds.
166
temperature field in the region close to the transiently developing jet. Moreover, the
circulation cell is not able to develop temperature contours that follow the velocity
contours in the circulation cell. However, radial extension of temperature contours are
observed for mean velocity of steady jet ≥ 100 m/s, i.e., the influence of streamline
curvature effect of the steady impinging jet on the convective heat transfer becomes
significant.
Figure 5.49 shows pressure contours for different entering mean velocity of the
steady impinging jet at 192.30 µs. The variation in size and orientation of the cir-
culation cell next to the jet boundary is evident with increasing mean velocity of
the steady jet. Moreover, stagnation region where both jets meet moves towards the
transiently developing jet when steady jet velocity increases. The radial expansion of
the transiently developing jets results in high-pressure region extending in the radial
direction.
Figure 5.50 shows the turbulence kinetic energy for different entering mean ve-
locity of the steady jet at different periods. The maximum turbulence kinetic energy
attains high values as mean steady jet velocity increases. Moreover, the maximum
turbulence kinetic energy is generated in the region next to the stagnation zone; which
is also confirmed in the previous study [10]. The location of the maximum turbulence
kinetic energy moves towards the transiently developing jet as mean velocity of the
steady jet increases. This is because of the stagnation region which moves towards
the transiently developing jet. This can also be seen from Figure 5.51, in which axial
velocity along the axial direction is shown, i.e., the location of zero axial velocity
167
Axial Distance (m)
Rad
ialD
ista
nce
(m)
101210
101220
101250
101240
101200
1012
45
0.000 0.001 0.002
0.000
0.001
0.002
0.003
Airjet velocity5 m/s
101230
101220
101200
1011
80
1011
30
101230101600101330
102170
0.000 0.001 0.002
0.000
0.001
0.002
0.003
Airjet velocity50 m/s
101180
1011
501010
95
101330101880
103300
100865
100930
0.000 0.001 0.002
0.000
0.001
0.002
0.003
Airjet velocity100 m/s
101130
101095
100930
1007
70
101180101330
101880
103300110350
101150
101140
107125
0.000 0.001 0.002
0.000
0.001
0.002
0.003
Airjet velocity150 m/s
Figure 5.49: Pressure (Pa) contours for four air jet velocities at 192.30 microsec-onds.
Figure 5.52: Variation of mass fraction of helium, nitrogen and oxygen in He-airmixture for four air jet velocites along the jet symmetry axis at r =0 m and t = 192.30 microseconds.
Figure 5.53: Temporal variation of mass fraction of helium, nitrogen and oxygenin He-air mixture along the jet symmetry axis at r = 0 m and at airjet velocity of 100 m/s.
173
Table 5.1: Thermophysical properties of fluids used in the simulations.
Transiently developing jet emanating from a free surface and expanding into air
ambient is considered to resemble the vapor jet behavior ejected from the laser-
produced cavity. Since the thermophysical properties of laser produced vapor is not
known, air or helium at 1500 K is considered as emerging jet in the simulations.
The jet exiting velocity profiles (jet velocity profiles onset of exiting the cavity) em-
ployed are obtained from the previous experimental study. This enables us to simulate
the actual laser produced cavity exiting conditions. A numerical scheme employing
control volume approach is employed when simulating the flow situations. The low
Reynolds number k − ε turbulence model is employed to account for the turbulence.
6.1.1 Transient Air Jet into Stagnant Air
It is found that in the early period the transient air jet expands more radially
than it does axially. As the time progresses, the jet width to jet penetration depth
ratio reduces. Once the jet exit velocity profiles become almost similar, self-similar
transient jet behavior is resulted. The specific conclusions derived from the present
174
175
study can be listed as follows:
1) Changes in jet exit velocity profiles result in variation in velocity magnitude
along the symmetry axis. This, in turn, alters the turbulence kinetic energy generation
in this region. Moreover, very small change in velocity magnitude results in large
change in turbulence kinetic energy.
2) In the early period temperature profiles follow the velocity profiles and the
convective heat transfer from the jet surface to its ambient is small. This occurs
because of small area of jet surface and short period of time, i.e., jet does not expand
enough to generate a large surface area and the period of expansion is short and hence
the heat transfer rate is low during early periods.
3) In the early period the ratio of jet expansion in the axial direction to the
one-fourth power of jet momentum rate per unit density increases linearly with the
square root of time, i.e., penetration number increases steadily. In this case, jet
behaves similar to those observed for slow flow jets. As time progresses, the rate of
momentum dissipation due to viscous dissipation becomes high and the penetration
number increases rapidly. This is observed for√t ≥ 12√µs in the present case.
6.1.2 Transient Helium Jet into Stagnant Air
It is found that the transient helium jet approaches self-similar behavior as the
dimensionless jet width (D/Zt) becomes 0.65 and the transition length of the jet is
obtained to be about 3 jet inlet diameters, which is less than the data reported in the
previous study. The specific conclusions derived from the present study are listed as
176
follows:
1) The flow entrainment results in a circulation cell in the region next to the jet
outer boundary after t = 76.92 µs. The orientation of circulation cell changes with
progressing time. This is due to the jet inlet velocity profiles, which varies with time.
2) Temperature contours follow almost the velocity profiles. Moreover, flow en-
trainment enhances the convective heat transfer rate, which in turn extends the high
temperature field into the jet-entrained region.
3) The mass ratio of helium decays sharply as the distance along the symmetry
axis increases away from the jet exit. The mixing of helium with air along the
symmetry axis indicates that while helium jet expands into the stagnant air, some
air molecules remain in the region close to the jet inlet as well as the expansion of
the helium into air accelerates the diffusional transport of air into helium jet.
4) In the early period of helium jet expansion, self-similar behavior of the jet is
not observed. This is because of the vast change of jet inlet velocity profiles in the
early period.
5) The constant penetration number
Ztµ •MHe/ρ
¶1/4×√t
is almost impossible dur-
ing the jet expansion in the present situation, since the jet inlet velocity changes
drastically in the early period.
6.2 Opposing Jets
The flow field due to opposing jet and transiently developing jet is studied in
relation to laser-induced ablation process. Steady jet represents the assisting gas
177
while transiently developing jet resembles the vapor plume emanating from the solid
surface during the ablation process. Since the thermophysical properties of the vapor
plume are not known, helium at 1500 K is employed for the transiently developing jet
while air is considered as an assisting gas. The flow, temperature and mass fraction
fields are simulated numerically using a control volume approach. The low Reynolds
number k − ε turbulence model is employed to account for the turbulence.
6.2.1 Transient Effect on The Flow Field Due to Opposing
Jets
It is found that the flow field in the region close to the transiently developing
jet is influenced considerably by the assisting gas jet. In the early stage transiently
developing jet expands in the axial direction and as the time progresses radial ex-
pansion of the jet dominates; in which case, a circulation cell next to the steady jet
boundary is developed. The specific conclusions derived from the present study can
be listed as follows:
1) The orientation and the size of the circulation cell next to the steady jet
boundary are influenced by the transiently developing jet entering velocities. In this
case, the velocity profile (similar to the profile for the fully developed flow) enhances
the size of the circulation cell.
2) The transiently developing jet influences the streamline curvature of the steady
impinging jet. In this case, enhancing radial flow suppresses the transiently developing
178
jet expansion in the axial direction.
3) The deep penetration of oxygen/nitrogen mass fraction into the region next
to the helium jet entry indicates that the flow mixing in this region occurs, i.e., next
to the stagnation region where two jets meet, the flow mixing is enhanced which can
also be observed through temperature contours. This is more pronounced as time
progresses.
4) A radial jet developed due to opposing of transiently developing and steady
jets. In this case, the radial jet show similar behavior to that corresponding to the
free jet, and the transiently developing jet characteristics do not affect considerably
the radial free jet characteristics.
5) The transiently developing jet does not approach a self-similar condition at
any period considered. Consequently, developing jet shows transient characteristics
for all the durations considered in the present study.
6) The penetration number of transiently developing jet attains almost steady
values during 10− 15√µs period. Moreover, sharp increase in penetration number in
the early period corresponds to the axial penetration of the jet.
6.2.2 Influence of Assisting Gas Velocity on The Flow Field
Due to Opposing Jets
It is found that the influence of the magnitude of the mean velocity of the steady
jet on the flow field in the region close to the transiently developing jet is found to be
179
very significant. In this case increasing mean velocity of impinging steady jet results
in development of the circulation cell next to steady jet boundary. The orientation
and strength of the circulation are modified by the mean velocity of the steady jet.
In this case, streamline curvature effect and flow entrainment cause flow mixing next
to the transiently developing jet boundary. Moreover, increasing mean velocity of the
steady jet suppresses the axial expansion of the transiently developing jet; hence, the
radial expansion of the jet is enhanced.
Mass fraction of helium decreases sharply in the axial direction as mean velocity
of the steady jet increases. Although the mass fraction ratio of O2/N2 in steady air
jet is 0.29 at jet inlet to the solution domain, it is modified in the region close to
the stagnation zone. In this case, oxygen mass fraction enhances in this region. This
suggests that flow mixing next to the transiently developing jet and diffusional mass
transport modify the mass fraction of species considerably in this region.
6.3 Future Work and Recommendations
Since laser-machining applications are involved with the laser non-conduction lim-
ited heating situation, phase change in the solid substrate should be considered when
modeling the laser-workpiece interaction. Moreover, the material properties vary with
temperature; therefore, temperature dependent properties should be accommodated
in the analysis. The rate of evaporation from the surface and the magnitude of recoil
pressure in the cavity determine the evaporating front velocity. Consequently, evapo-
rating front velocity of the vapor jet emanating from the cavity needs to be considered
180
when simulating the impinging gas effects on the laser-workpiece interaction.
Although the thermophysical properties of the evaporating front are not known,
a high-density gas, such as steam, can be introduced in the simulations to resemble
the evaporating front. The nucleation formation in the superheated liquid in the
cavity and droplet formation in the vapor phase during the plume expansion could
be accommodated in the heating model. Moreover, the influence of retarding zone on
the vapor front expansion could be included in the analysis.
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